2007
2007
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Paper 1, Section I,
Part IA, 2007 comment(i) The spherical polar unit basis vectors and in are given in terms of the Cartesian unit basis vectors and by
Express and in terms of and .
(ii) Use suffix notation to prove the following identity for the vectors , and in :
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Paper 1, Section I, B
Part IA, 2007 commentFor the equations
find the values of and for which
(i) there is a unique solution;
(ii) there are infinitely many solutions;
(iii) there is no solution.
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Paper 1, Section II, C
Part IA, 2007 comment(i) Describe geometrically the following surfaces in three-dimensional space:
(a) , where
(b) , where .
Here and are fixed scalars and is a fixed unit vector. You should identify the meaning of and for these surfaces.
(ii) The plane , where is a fixed unit vector, and the sphere with centre and radius intersect in a circle with centre and radius .
(a) Show that , where you should give in terms of and .
(b) Find in terms of and .
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Paper 1, Section II, C
Part IA, 2007 commentLet be the linear map defined by
where and are positive scalar constants, and is a unit vector.
(i) By considering the effect of on and on a vector orthogonal to , describe geometrically the action of .
(ii) Express the map as a matrix using suffix notation. Find and in the case
(iii) Find, in the general case, the inverse map (i.e. express in terms of in vector form).
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Paper 1, Section II, C
Part IA, 2007 commentLet and be non-zero vectors in a real vector space with scalar product denoted by . Prove that , and prove also that if and only if for some scalar .
(i) By considering suitable vectors in , or otherwise, prove that the inequality holds for any real numbers and .
(ii) By considering suitable vectors in , or otherwise, show that only one choice of real numbers satisfies , and find these numbers.
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Paper 1, Section II, A
Part IA, 2007 comment(i) Show that any line in the complex plane can be represented in the form
where and .
(ii) If and are two complex numbers for which
show that either or is real.
(iii) Show that any Möbius transformation
that maps the real axis into the unit circle can be expressed in the form
where and .
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Paper 3, Section I, D
Part IA, 2007 commentProve that every permutation of may be expressed as a product of disjoint cycles.
Let and let . Write as a product of disjoint cycles. What is the order of
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Paper 3, Section I, D
Part IA, 2007 commentWhat does it mean to say that groups and are isomorphic?
Prove that no two of and are isomorphic. [Here denotes the cyclic group of order .]
Give, with justification, a group of order 8 that is not isomorphic to any of those three groups.
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Paper 3, Section II, D
Part IA, 2007 commentLet be a real symmetric matrix. Prove that every eigenvalue of is real, and that eigenvectors corresponding to distinct eigenvalues are orthogonal. Indicate clearly where in your argument you have used the fact that is real.
What does it mean to say that a real matrix is orthogonal ? Show that if is orthogonal and is as above then is symmetric. If is any real invertible matrix, must be symmetric? Justify your answer.
Give, with justification, real matrices with the following properties:
(i) has no real eigenvalues;
(ii) is not diagonalisable over ;
(iii) is diagonalisable over , but not over ;
(iv) is diagonalisable over , but does not have an orthonormal basis of eigenvectors.
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Paper 3, Section II, D
Part IA, 2007 commentIn the group of Möbius maps, what is the order of the Möbius map ? What is the order of the Möbius map ?
Prove that every Möbius map is conjugate either to a map of the form (some ) or to the . Is conjugate to a map of the form
Let be a Möbius map of order , for some positive integer . Under the action on of the group generated by , what are the various sizes of the orbits? Justify your answer.
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Paper 3, Section II, D
Part IA, 2007 commentLet be an element of a finite group . What is meant by the order of ? Prove that the order of must divide the order of . [No version of Lagrange's theorem or the Orbit-Stabilizer theorem may be used without proof.]
If is a group of order , and is a divisor of with , is it always true that must contain an element of order ? Justify your answer.
Prove that if and are coprime then the group is cyclic.
If and are not coprime, can it happen that is cyclic?
[Here denotes the cyclic group of order .]
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Paper 3, Section II,
Part IA, 2007 commentWhat does it mean to say that a subgroup of a group is normal? Give, with justification, an example of a subgroup of a group that is normal, and also an example of a subgroup of a group that is not normal.
If is a normal subgroup of , explain carefully how to make the set of (left) cosets of into a group.
Let be a normal subgroup of a finite group . Which of the following are always true, and which can be false? Give proofs or counterexamples as appropriate.
(i) If is cyclic then and are cyclic.
(ii) If and are cyclic then is cyclic.
(iii) If is abelian then and are abelian.
(iv) If and are abelian then is abelian.
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Paper 1, Section I,
Part IA, 2007 commentProve that, for positive real numbers and ,
For positive real numbers , prove that the convergence of
implies the convergence of
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Paper 1, Section I, D
Part IA, 2007 commentLet be a complex power series. Show that there exists such that converges whenever and diverges whenever .
Find the value of for the power series
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Paper 1, Section II, F
Part IA, 2007 commentLet , and consider the sequence of positive real numbers defined by
Show that for all . Prove that the sequence converges to a limit.
Suppose instead that . Prove that again the sequence converges to a limit.
Prove that the limits obtained in the two cases are equal.
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Paper 1, Section II, E
Part IA, 2007 commentState and prove the Mean Value Theorem.
Let be a function such that, for every exists and is non-negative.
(i) Show that if then .
(ii) Let and . Show that there exist and such that
and that
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Paper 1, Section II, E
Part IA, 2007 commentLet be real numbers, and let be continuous. Show that is bounded on , and that there exist such that for all , .
Let be a continuous function such that
Show that is bounded. Show also that, if and are real numbers with , then there exists with .
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Paper 1, Section II, D
Part IA, 2007 commentExplain carefully what it means to say that a bounded function is Riemann integrable.
Prove that every continuous function is Riemann integrable.
For each of the following functions from to , determine with proof whether or not it is Riemann integrable:
(i) the function for , with ;
(ii) the function for , with .
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Paper 2, Section I, B
Part IA, 2007 commentFind the solution of the equation
that satisfies and .
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Paper 2, Section I, B
Part IA, 2007 commentInvestigate the stability of:
(i) the equilibrium points of the equation
(ii) the constant solutions of the discrete equation
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Paper 2, Section II, B
Part IA, 2007 comment(i) The function satisfies the equation
Give the definitions of the terms ordinary point, singular point, and regular singular point for this equation.
(ii) For the equation
classify the point according to the definitions you gave in (i), and find the series solutions about . Identify these solutions in closed form.
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Paper 2, Section II, B
Part IA, 2007 commentFind the most general solution of the equation
by making the change of variables
Find the solution that satisfies and when .
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Paper 2, Section II, B
Part IA, 2007 comment(i) Find, in the form of an integral, the solution of the equation
that satisfies as . Here is a general function and is a positive constant.
Hence find the solution in each of the cases:
(a) ;
(b) , where is the Heaviside step function.
(ii) Find and sketch the solution of the equation
given that and is continuous.
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Paper 2, Section II, B
Part IA, 2007 comment(i) Find the general solution of the difference equation
(ii) Find the solution of the equation
that satisfies . Hence show that, for any positive integer , the quantity is divisible by
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Paper 4, Section I, C
Part IA, 2007 commentA rocket, moving vertically upwards, ejects gas vertically downwards at speed relative to the rocket. Derive, giving careful explanations, the equation of motion
where and are the speed and total mass of the rocket (including fuel) at time .
If is constant and the rocket starts from rest with total mass , show that
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Paper 4, Section I, C
Part IA, 2007 commentSketch the graph of .
A particle of unit mass moves along the axis in the potential . Sketch the phase plane, and describe briefly the motion of the particle on the different trajectories.
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Paper 4, Section II, C
Part IA, 2007 commentA small ring of mass is threaded on a smooth rigid wire in the shape of a parabola given by , where measures horizontal distance and measures distance vertically upwards. The ring is held at height , then released.
(i) Show by dimensional analysis that the period of oscillations, , can be written in the form
for some function .
(ii) Show that is given by
and find, to first order in , the period of small oscillations.
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Paper 4, Section II, C
Part IA, 2007 commentA particle of mass experiences, at the point with position vector , a force given by
where and are positive constants and is a constant, uniform, vector field.
(i) Show that is constant. Give a physical interpretation of each term and a physical explanation of the fact that does not arise in this expression.
(ii) Show that is constant.
(iii) Given that the particle was initially at rest at , derive an expression for at time .
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Paper 4, Section II, C
Part IA, 2007 commentA particle moves in the gravitational field of the Sun. The angular momentum per unit mass of the particle is and the mass of the Sun is . Assuming that the particle moves in a plane, write down the equations of motion in polar coordinates, and derive the equation
where and .
Write down the equation of the orbit ( as a function of ), given that the particle moves with the escape velocity and is at the perihelion of its orbit, a distance from the Sun, when . Show that
and hence that the particle reaches a distance from the Sun at time .
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Paper 4, Section II,
Part IA, 2007 commentThe th particle of a system of particles has mass and, at time , position vector with respect to an origin . It experiences an external force , and also an internal force due to the th particle (for each ), where is parallel to and Newton's third law applies.
(i) Show that the position of the centre of mass, , satisfies
where is the total mass of the system and is the sum of the external forces.
(ii) Show that the total angular momentum of the system about the origin, , satisfies
where is the total moment about the origin of the external forces.
(iii) Show that can be expressed in the form
where is the velocity of the centre of mass, is the position vector of the th particle relative to the centre of mass, and is the velocity of the th particle relative to the centre of mass.
(iv) In the case when the internal forces are derived from a potential , where , and there are no external forces, show that
where is the total kinetic energy of the system.
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Paper 4, Section I,
Part IA, 2007 comment(i) Use Euclid's algorithm to find all pairs of integers and such that
(ii) Show that, if is odd, then is divisible by 24 .
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Paper 4, Section I,
Part IA, 2007 commentFor integers and with , define . Arguing from your definition, show that
for all integers and with .
Use induction on to prove that
for all non-negative integers and .
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Paper 4 , Section II, E
Part IA, 2007 commentState and prove the Inclusion-Exclusion principle.
The keypad on a cash dispenser is broken. To withdraw money, a customer is required to key in a 4-digit number. However, the key numbered 0 will only function if either the immediately preceding two keypresses were both 1 , or the very first key pressed was 2. Explaining your reasoning clearly, use the Inclusion-Exclusion Principle to find the number of 4-digit codes which can be entered.
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Paper 4, Section II,
Part IA, 2007 commentStating carefully any results about countability you use, show that for any the set of polynomials with integer coefficients in variables is countable. By taking , deduce that there exist uncountably many transcendental numbers.
Show that there exists a sequence of real numbers with the property that for every and for every non-zero polynomial .
[You may assume without proof that is uncountable.]
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Paper 4, Section II,
Part IA, 2007 commentLet be real numbers.
What does it mean to say that the sequence converges?
What does it mean to say that the series converges?
Show that if is convergent, then . Show that the converse can be false.
Sequences of positive real numbers are given, such that the inequality
holds for all . Show that, if diverges, then .
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Paper 4, Section II, E
Part IA, 2007 comment(i) Let be a prime number, and let and be integers such that divides . Show that at least one of and is divisible by . Explain how this enables one to prove the Fundamental Theorem of Arithmetic.
[Standard properties of highest common factors may be assumed without proof.]
(ii) State and prove the Fermat-Euler Theorem.
Let have decimal expansion with . Use the fact that to show that, for every .
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Paper 2, Section I, F
Part IA, 2007 commentLet and be independent random variables, each uniformly distributed on . Let and . Show that , and hence find the covariance of and .
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Paper 2, Section I, F
Part IA, 2007 commentLet be a normally distributed random variable with mean 0 and variance 1 . Define, and determine, the moment generating function of . Compute for .
Let be a normally distributed random variable with mean and variance . Determine the moment generating function of .
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Paper 2, Section II, F
Part IA, 2007 commentLet be a non-negative integer-valued random variable with
Define , and show that
Let be a sequence of independent and identically distributed continuous random variables. Let the random variable mark the point at which the sequence stops decreasing: that is, is such that
where, if there is no such finite value of , we set . Compute , and show that . Determine .
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Paper 2, Section II, F
Part IA, 2007 commentLet and be independent non-negative random variables, with densities and respectively. Find the joint density of and , where is a positive constant.
Let and be independent and exponentially distributed random variables, each with density
Find the density of . Is it the same as the density of the random variable
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Paper 2, Section II, F
Part IA, 2007 commentLet be events in a sample space. For each of the following statements, either prove the statement or provide a counterexample.
(i)
(ii)
(iii)
(iv) If is an event and if, for each is a pair of independent events, then is also a pair of independent events.
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Paper 2, Section II, F
Part IA, 2007 commentLet and be three random points on a sphere with centre . The positions of and are independent, and each is uniformly distributed over the surface of the sphere. Calculate the probability density function of the angle formed by the lines and .
Calculate the probability that all three of the angles and are acute. [Hint: Condition on the value of the angle .]
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Paper 3 , Section I, A
Part IA, 2007 comment(i) Give definitions for the unit tangent vector and the curvature of a parametrised curve in . Calculate and for the circular helix
where and are constants.
(ii) Find the normal vector and the equation of the tangent plane to the surface in given by
at the point .
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Paper 3, Section I, A
Part IA, 2007 commentBy using suffix notation, prove the following identities for the vector fields and B in :
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Paper 3, Section II, A
Part IA, 2007 comment(i) Define what is meant by a conservative vector field. Given a vector field and a function defined in , show that, if is a conservative vector field, then
(ii) Given two functions and defined in , prove Green's theorem,
where is a simple closed curve bounding a region in .
Through an appropriate choice for and , find an expression for the area of the region , and apply this to evaluate the area of the ellipse bounded by the curve
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Paper 3, Section II, A
Part IA, 2007 commentFor a given charge distribution and divergence-free current distribution (i.e. in , the electric and magnetic fields and satisfy the equations
The radiation flux vector is defined by . For a closed surface around a region , show using Gauss' theorem that the flux of the vector through can be expressed as
For electric and magnetic fields given by
find the radiation flux through the quadrant of the unit spherical shell given by
[If you use (*), note that an open surface has been specified.]
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Paper 3, Section II, A
Part IA, 2007 commentThe function satisfies in and on , where is a region of which is bounded by the surface . Prove that everywhere in .
Deduce that there is at most one function satisfying in and on , where and are given functions.
Given that the function depends only on the radial coordinate , use Cartesian coordinates to show that
Find the general solution in this radial case for where is a constant.
Find solutions for a solid sphere of radius with a central cavity of radius in the following three regions:
(i) where and and bounded as ;
(ii) where and ;
(iii) where and and as .
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Paper 3, Section II, A
Part IA, 2007 commentShow that any second rank Cartesian tensor in can be written as a sum of a symmetric tensor and an antisymmetric tensor. Further, show that can be decomposed into the following terms
where is symmetric and traceless. Give expressions for and explicitly in terms of .
For an isotropic material, the stress can be related to the strain through the stress-strain relation, , where the elasticity tensor is given by
and and are scalars. As in , the strain can be decomposed into its trace , a symmetric traceless tensor and a vector . Use the stress-strain relation to express each of and in terms of and .
Hence, or otherwise, show that if is symmetric then so is . Show also that the stress-strain relation can be written in the form
where and are scalars.
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1.I.1G
Part IB, 2007 commentSuppose that is a basis of the complex vector space and that is the linear operator defined by , and .
By considering the action of on column vectors of the form , where , or otherwise, find the diagonalization of and its characteristic polynomial.
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1.II.9G
Part IB, 2007 commentState and prove Sylvester's law of inertia for a real quadratic form.
[You may assume that for each real symmetric matrix A there is an orthogonal matrix , such that is diagonal.]
Suppose that is a real vector space of even dimension , that is a non-singular quadratic form on and that is an -dimensional subspace of on which vanishes. What is the signature of
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2.I.1G
Part IB, 2007 commentSuppose that are endomorphisms of the 3-dimensional complex vector space and that the eigenvalues of each of them are . What are their characteristic and minimal polynomials? Are they conjugate?
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2.II.10G
Part IB, 2007 commentSuppose that is the complex vector space of complex polynomials in one variable, .
(i) Show that the form , defined by
is a positive definite Hermitian form on .
(ii) Find an orthonormal basis of for this form, in terms of the powers of .
(iii) Generalize this construction to complex vector spaces of complex polynomials in any finite number of variables.
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3.II.10G
Part IB, 2007 comment(i) Define the terms row-rank, column-rank and rank of a matrix, and state a relation between them.
(ii) Fix positive integers with . Suppose that is an matrix and a matrix. State and prove the best possible upper bound on the rank of the product .
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4.I.1G
Part IB, 2007 commentSuppose that is a linear map of finite-dimensional complex vector spaces. What is the dual map of the dual vector spaces?
Suppose that we choose bases of and take the corresponding dual bases of the dual vector spaces. What is the relation between the matrices that represent and with respect to these bases? Justify your answer.
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4.II.10G
Part IB, 2007 comment(i) State and prove the Cayley-Hamilton theorem for square complex matrices.
(ii) A square matrix is of order for a strictly positive integer if and no smaller positive power of is equal to .
Determine the order of a complex matrix of trace zero and determinant 1 .
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1.II.10G
Part IB, 2007 comment(i) State a structure theorem for finitely generated abelian groups.
(ii) If is a field and a polynomial of degree in one variable over , what is the maximal number of zeroes of ? Justify your answer in terms of unique factorization in some polynomial ring, or otherwise.
(iii) Show that any finite subgroup of the multiplicative group of non-zero elements of a field is cyclic. Is this true if the subgroup is allowed to be infinite?
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3.I.1G
Part IB, 2007 commentWhat are the orders of the groups and where is the field of elements?
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3.II.11G
Part IB, 2007 comment(i) State the Sylow theorems for Sylow -subgroups of a finite group.
(ii) Write down one Sylow 3-subgroup of the symmetric group on 5 letters. Calculate the number of Sylow 3-subgroups of .
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2.I.2G
Part IB, 2007 commentDefine the term Euclidean domain.
Show that the ring of integers is a Euclidean domain.
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2.II.11G
Part IB, 2007 comment(i) Give an example of a Noetherian ring and of a ring that is not Noetherian. Justify your answers.
(ii) State and prove Hilbert's basis theorem.
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4.I.2G
Part IB, 2007 commentIf is a prime, how many abelian groups of order are there, up to isomorphism?
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4.II.11G
Part IB, 2007 commentA regular icosahedron has 20 faces, 12 vertices and 30 edges. The group of its rotations acts transitively on the set of faces, on the set of vertices and on the set of edges.
(i) List the conjugacy classes in and give the size of each.
(ii) Find the order of and list its normal subgroups.
[A normal subgroup of is a union of conjugacy classes in .]
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1.I.2A
Part IB, 2007 commentState the Gauss-Bonnet theorem for spherical triangles, and deduce from it that for each convex polyhedron with faces, edges, and vertices, .
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2.II.12A
Part IB, 2007 comment(i) The spherical circle with centre and radius , is the set of all points on the unit sphere at spherical distance from . Find the circumference of a spherical circle with spherical radius . Compare, for small , with the formula for a Euclidean circle and comment on the result.
(ii) The cross ratio of four distinct points in is
Show that the cross-ratio is a real number if and only if lie on a circle or a line.
[You may assume that Möbius transformations preserve the cross-ratio.]
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3.I
Part IB, 2007 commentLet be a line in the Euclidean plane and a point on . Denote by the reflection in and by the rotation through an angle about . Describe, in terms of , and , a line fixed by the composition and show that is a reflection.
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3.II.12A
Part IB, 2007 commentFor a parameterized smooth embedded surface , where is an open domain in , define the first fundamental form, the second fundamental form, and the Gaussian curvature . State the Gauss-Bonnet formula for a compact embedded surface having Euler number .
Let denote a surface defined by rotating a curve
about the -axis. Here are positive constants, such that and . By considering a smooth parameterization, find the first fundamental form and the second fundamental form of .
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4.II.12A
Part IB, 2007 commentWrite down the Riemannian metric for the upper half-plane model of the hyperbolic plane. Describe, without proof, the group of isometries of and the hyperbolic lines (i.e. the geodesics) on .
Show that for any two hyperbolic lines , there is an isometry of which maps onto .
Suppose that is a composition of two reflections in hyperbolic lines which are ultraparallel (i.e. do not meet either in the hyperbolic plane or at its boundary). Show that cannot be an element of finite order in the group of isometries of .
[Existence of a common perpendicular to two ultraparallel hyperbolic lines may be assumed. You might like to choose carefully which hyperbolic line to consider as a common perpendicular.]
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1.II.11H
Part IB, 2007 commentDefine what it means for a function to be differentiable at a point with derivative a linear map
State the Chain Rule for differentiable maps and . Prove the Chain Rule.
Let denote the standard Euclidean norm of . Find the partial derivatives of the function where they exist.
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Part IB, 2007
commentFor integers and , define to be 0 if , or if and is the largest non-negative integer such that is a multiple of . Show that is a metric on the integers .
Does the sequence converge in this metric?
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2.II.13H
Part IB, 2007 commentShow that the limit of a uniformly convergent sequence of real valued continuous functions on is continuous on .
Let be a sequence of continuous functions on which converge point-wise to a continuous function. Suppose also that the integrals converge to . Must the functions converge uniformly to Prove or give a counterexample.
Let be a sequence of continuous functions on which converge point-wise to a function . Suppose that is integrable and that the integrals converge to . Is the limit necessarily continuous? Prove or give a counterexample.
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Part IB, 2007
commentDefine uniform continuity for a real-valued function on an interval in the real line. Is a uniformly continuous function on the real line necessarily bounded?
Which of the following functions are uniformly continuous on the real line?
(i) ,
(ii) .
Justify your answers.
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Part IB, 2007
commentLet be the real vector space of continuous functions . Show that defining
makes a normed vector space.
Define for positive integers . Is the sequence convergent to some element of ? Is a Cauchy sequence in ? Justify your answers.
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4.I.3H
Part IB, 2007 commentDefine uniform convergence for a sequence of real-valued functions on the interval .
For each of the following sequences of functions on , find the pointwise limit function. Which of these sequences converge uniformly on ?
(i) ,
(ii) .
Justify your answers.
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4.II.13H
Part IB, 2007 commentState and prove the Contraction Mapping Theorem.
Find numbers and , with , such that the mapping defined by
is a contraction, in the sup norm on . Deduce that the differential equation
has a unique solution in some interval containing 0 .
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1.II.12A
Part IB, 2007 commentLet and be topological spaces. Define the product topology on and show that if and are Hausdorff then so is .
Show that the following statements are equivalent.
(i) is a Hausdorff space.
(ii) The diagonal is a closed subset of , in the product topology.
(iii) For any topological space and any continuous maps , the set is closed in .
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2.I.4A
Part IB, 2007 commentAre the following statements true or false? Give a proof or a counterexample as appropriate.
(i) If is a continuous map of topological spaces and is compact then is compact.
(ii) If is a continuous map of topological spaces and is compact then is compact.
(iii) If a metric space is complete and a metric space is homeomorphic to then is complete.
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3.I.4A
Part IB, 2007 comment(a) Let be a connected topological space such that each point of has a neighbourhood homeomorphic to . Prove that is path-connected.
(b) Let denote the topology on , such that the open sets are , the empty set, and all the sets , for . Prove that any continuous map from the topological space to the Euclidean is constant.
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4.II.14A
Part IB, 2007 comment(a) For a subset of a topological space , define the closure cl of . Let be a map to a topological space . Prove that is continuous if and only if , for each .
(b) Let be a metric space. A subset of is called dense in if the closure of is equal to .
Prove that if a metric space is compact then it has a countable subset which is dense in .
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1.I.3F
Part IB, 2007 commentFor the function
determine the Taylor series of around the point , and give the largest for which this series converges in the disc .
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1.II.13F
Part IB, 2007 commentBy integrating round the contour , which is the boundary of the domain
evaluate each of the integrals
[You may use the relations and for
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2.II.14F
Part IB, 2007 commentLet be the half-strip in the complex plane,
Find a conformal mapping that maps onto the unit disc.
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3.II.14H
Part IB, 2007 commentSay that a function on the complex plane is periodic if and for all . If is a periodic analytic function, show that is constant.
If is a meromorphic periodic function, show that the number of zeros of in the square is equal to the number of poles, both counted with multiplicities.
Define
where the sum runs over all with and integers, not both 0 . Show that this series converges to a meromorphic periodic function on the complex plane.
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4.I.4H
Part IB, 2007 commentState the argument principle.
Show that if is an analytic function on an open set which is one-to-one, then for all .
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Part IB, 2007
commentShow that the function is harmonic. Find its harmonic conjugate and the analytic function whose real part is . Sketch the curves and .
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4.II.15F
Part IB, 2007 comment(i) Use the definition of the Laplace transform of :
to show that, for ,
(ii) Use contour integration to find the inverse Laplace transform of
(iii) Verify the result in (ii) by using the results in (i) and the convolution theorem.
(iv) Use Laplace transforms to solve the differential equation
subject to the initial conditions
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1.II.14D
Part IB, 2007 commentDefine the Fourier transform of a function that tends to zero as , and state the inversion theorem. State and prove the convolution theorem.
Calculate the Fourier transforms of
Hence show that
and evaluate this integral for all other (real) values of .
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Part IB, 2007
commentShow that a smooth function that satisfies can be written as a Fourier series of the form
where the should be specified. Write down an integral expression for .
Hence solve the following differential equation
with boundary conditions , in the form of an infinite series.
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2.II.15D
Part IB, 2007 commentLet be a non-zero solution of the Sturm-Liouville equation
with boundary conditions . Show that, if and are related by
with satisfying the same boundary conditions as , then
Suppose that is normalised so that
and consider the problem
By choosing appropriately in deduce that, if
then
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Part IB, 2007
commentDescribe the method of Lagrange multipliers for finding extrema of a function subject to the constraint that .
Illustrate the method by finding the maximum and minimum values of for points lying on the ellipsoid
with and all positive.
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3.II.15E
Part IB, 2007 commentLegendre's equation may be written
Show that if is a positive integer, this equation has a solution that is a polynomial of degree . Find and explicitly.
Write down a general separable solution of Laplace's equation, , in spherical polar coordinates . (A derivation of this result is not required.)
Hence or otherwise find when
with both when and when .
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4.I.5B
Part IB, 2007 commentShow that the general solution of the wave equation
where is a constant, is
where and are twice differentiable functions. Briefly discuss the physical interpretation of this solution.
Calculate subject to the initial conditions
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4.II.16E
Part IB, 2007 commentWrite down the Euler-Lagrange equation for extrema of the functional
Show that a first integral of this equation is given by
A road is built between two points and in the plane whose polar coordinates are and respectively. Owing to congestion, the traffic speed at points along the road is with a positive constant. If the equation describing the road is , obtain an integral expression for the total travel time from to .
[Arc length in polar coordinates is given by .]
Calculate for the circular road .
Find the equation for the road that minimises and determine this minimum value.
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1.II.15B
Part IB, 2007 commentThe relative motion of a neutron and proton is described by the Schrödinger equation for a single particle of mass under the influence of the central potential
where and are positive constants. Solve this equation for a spherically symmetric state of the deuteron, which is a bound state of a proton and neutron, giving the condition on for this state to exist.
[If is spherically symmetric then
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2.II.16B
Part IB, 2007 commentWrite down the angular momentum operators in terms of the position and momentum operators, and , and the commutation relations satisfied by and .
Verify the commutation relations
Further, show that
A wave-function is spherically symmetric. Verify that
Consider the vector function . Show that and are eigenfunctions of with eigenvalues respectively.
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3.I.7B
Part IB, 2007 commentThe quantum mechanical harmonic oscillator has Hamiltonian
and is in a stationary state of energy . Show that
where and . Use the Heisenberg Uncertainty Principle to show that
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3.II.16B
Part IB, 2007 commentA quantum system has a complete set of orthonormal eigenstates, , with nondegenerate energy eigenvalues, , where Write down the wave-function, in terms of the eigenstates.
A linear operator acts on the system such that
Find the eigenvalues of and obtain a complete set of normalised eigenfunctions, , of in terms of the .
At time a measurement is made and it is found that the observable corresponding to has value 3. After time is measured again. What is the probability that the value is found to be 1 ?
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4.I.6B
Part IB, 2007 commentA particle moving in one space dimension with wave-function obeys the time-dependent Schrödinger equation. Write down the probability density, , and current density, , in terms of the wave-function and show that they obey the equation
The wave-function is
where and is a constant, which may be complex. Evaluate .
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1.II.16E
Part IB, 2007 commentA steady magnetic field is generated by a current distribution that vanishes outside a bounded region . Use the divergence theorem to show that
Define the magnetic vector potential . Use Maxwell's equations to obtain a differential equation for in terms of .
It may be shown that for an unbounded domain the equation for has solution
Deduce that in general the leading order approximation for as is
Find the corresponding far-field expression for .
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Part IB, 2007
commentA metal has uniform conductivity . A cylindrical wire with radius and length is manufactured from the metal. Show, using Maxwell's equations, that when a steady current flows along the wire the current density within the wire is uniform.
Deduce the electrical resistance of the wire and the rate of Ohmic dissipation within it.
Indicate briefly, and without detailed calculation, whether your results would be affected if the wire was not straight.
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2.II.17E
Part IB, 2007 commentIf is a fixed surface enclosing a volume , use Maxwell's equations to show that
where . Give a physical interpretation of each term in this equation.
Show that Maxwell's equations for a vacuum permit plane wave solutions with with and constants, and determine the relationship between and .
Find also the corresponding and hence the time average . What does represent in this case?
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3.II.17E
Part IB, 2007 commentA capacitor consists of three long concentric cylinders of radii and respectively, where . The inner and outer cylinders are earthed (i.e. held at zero potential); the cylinder of radius is raised to a potential . Find the electrostatic potential in the regions between the cylinders and deduce the capacitance, per unit length, of the system.
For with find correct to leading order in and comment on your result.
Find also the value of at which has an extremum. Is the extremum a maximum or a minimum? Justify your answer.
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4.I
Part IB, 2007 commentWrite down Faraday's law of electromagnetic induction for a moving circuit in a magnetic field . Explain carefully the meaning of each term in the equation.
A thin wire is bent into a circular loop of radius . The loop lies in the -plane at time . It spins steadily with angular velocity , where is a constant and is a unit vector in the -direction. A spatially uniform magnetic field is applied, with and both constant. If the resistance of the wire is , find the current in the wire at time .
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1.I.4B
Part IB, 2007 commentWrite down the position four-vector. Suppose this represents the position of a particle with rest mass and velocity v. Show that the four momentum of the particle is
where .
For a particle of zero rest mass show that
where is the three momentum.
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2.I.7B
Part IB, 2007 commentA particle in inertial frame has coordinates , whilst the coordinates are in frame , which moves with relative velocity in the direction. What is the relationship between the coordinates of and ?
Frame , with cooordinates , moves with velocity with respect to and velocity with respect to . Derive the relativistic formula for in terms of and . Show how the Newtonian limit is recovered.
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4.II.17B
Part IB, 2007 comment(a) A moving particle of rest-mass decays into two photons of zero rest-mass,
Show that
where is the angle between the three-momenta of the two photons and are their energies.
(b) The particle of rest-mass decays into an electron of rest-mass and a neutrino of zero rest mass,
Show that , the speed of the electron in the rest frame of the , is
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1.I.5D
Part IB, 2007 commentA steady two-dimensional velocity field is given by
(i) Calculate the vorticity of the flow.
(ii) Verify that is a possible flow field for an incompressible fluid, and calculate the stream function.
(iii) Show that the streamlines are bounded if and only if .
(iv) What are the streamlines in the case
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1.II.17D
Part IB, 2007 commentWrite down the Euler equation for the steady motion of an inviscid, incompressible fluid in a constant gravitational field. From this equation, derive (a) Bernoulli's equation and (b) the integral form of the momentum equation for a fixed control volume with surface .
(i) A circular jet of water is projected vertically upwards with speed from a nozzle of cross-sectional area at height . Calculate how the speed and crosssectional area of the jet vary with , for .
(ii) A circular jet of speed and cross-sectional area impinges axisymmetrically on the vertex of a cone of semi-angle , spreading out to form an almost parallel-sided sheet on the surface. Choose a suitable control volume and, neglecting gravity, show that the force exerted by the jet on the cone is
(iii) A cone of mass is supported, axisymmetrically and vertex down, by the jet of part (i), with its vertex at height , where . Assuming that the result of part (ii) still holds, show that is given by
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2.I.8D
Part IB, 2007 commentAn incompressible, inviscid fluid occupies the region beneath the free surface and moves with a velocity field given by the velocity potential ; gravity acts in the direction. Derive the kinematic and dynamic boundary conditions that must be satisfied by on .
[You may assume Bernoulli's integral of the equation of motion:
In the absence of waves, the fluid has uniform velocity in the direction. Derive the linearised form of the above boundary conditions for small amplitude waves, and verify that they and Laplace's equation are satisfied by the velocity potential
where , with a corresponding expression for , as long as
What are the propagation speeds of waves with a given wave-number
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Part IB, 2007
commentGiven that the circulation round every closed material curve in an inviscid, incompressible fluid remains constant in time, show that the velocity field of such a fluid started from rest can be written as the gradient of a potential, , that satisfies Laplace's equation.
A rigid sphere of radius a moves in a straight line at speed in a fluid that is at rest at infinity. Using axisymmetric spherical polar coordinates , with in the direction of motion, write down the boundary conditions on and, by looking for a solution of the form , show that the velocity potential is given by
Calculate the kinetic energy of the fluid.
A rigid sphere of radius and uniform density is submerged in an infinite fluid of density , under the action of gravity. Show that, when the sphere is released from rest, its initial upwards acceleration is
[Laplace's equation for an axisymmetric scalar field in spherical polars is:
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4.II.18D
Part IB, 2007 commentStarting from Euler's equation for an inviscid, incompressible fluid in the absence of body forces,
derive the equation for the vorticity .
[You may assume that
Show that, in a two-dimensional flow, vortex lines keep their strength and move with the fluid.
Show that a two-dimensional flow driven by a line vortex of circulation at distance from a rigid plane wall is the same as if the wall were replaced by another vortex of circulation at the image point, distance from the wall on the other side. Deduce that the first vortex will move at speed parallel to the wall.
A line vortex of circulation moves in a quarter-plane, bounded by rigid plane walls at and . Show that the vortex follows a trajectory whose equation in plane polar coordinates is constant.
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1.I.6F
Part IB, 2007 commentSolve the least squares problem
using method with Householder transformation. (A solution using normal equations is not acceptable.)
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2.II.18F
Part IB, 2007 commentFor a symmetric, positive definite matrix with the spectral radius , the linear system is solved by the iterative procedure
where is a real parameter. Find the range of that guarantees convergence of to the exact solution for any choice of .
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3.II.19F
Part IB, 2007 commentProve that the monic polynomials , orthogonal with respect to a given weight function on , satisfy the three-term recurrence relation
where . Express the values and in terms of and and show that .
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4.I.8F
Part IB, 2007 commentGiven , we approximate by the linear combination
Using the Peano kernel theorem, determine the least constant in the inequality
and give an example of for which the inequality turns into equality.
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1.I.7C
Part IB, 2007 commentLet be independent, identically distributed random variables from the distribution where and are unknown. Use the generalized likelihood-ratio test to derive the form of a test of the hypothesis against .
Explain carefully how the test should be implemented.
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1.II.18C
Part IB, 2007 commentLet be independent, identically distributed random variables with
where is an unknown parameter, , and . It is desired to estimate the quantity .
(i) Find the maximum-likelihood estimate, , of .
(ii) Show that is an unbiased estimate of and hence, or otherwise, obtain an unbiased estimate of which has smaller variance than and which is a function of .
(iii) Now suppose that a Bayesian approach is adopted and that the prior distribution for , is taken to be the uniform distribution on . Compute the Bayes point estimate of when the loss function is .
[You may use that fact that when are non-negative integers,
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2.II.19C
Part IB, 2007 commentState and prove the Neyman-Pearson lemma.
Suppose that is a random variable drawn from the probability density function
where and is unknown. Find the most powerful test of size , , of the hypothesis against the alternative . Express the power of the test as a function of .
Is your test uniformly most powerful for testing against Explain your answer carefully.
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3.I.8C
Part IB, 2007 commentLight bulbs are sold in packets of 3 but some of the bulbs are defective. A sample of 256 packets yields the following figures for the number of defectives in a packet:
\begin{tabular}{l|cccc} No. of defectives & 0 & 1 & 2 & 3 \ \hline No. of packets & 116 & 94 & 40 & 6 \end{tabular}
Test the hypothesis that each bulb has a constant (but unknown) probability of being defective independently of all other bulbs.
[Hint: You may wish to use some of the following percentage points:
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4.II.19C
Part IB, 2007 commentConsider the linear regression model
where are independent, identically distributed are known real numbers with and and are unknown.
(i) Find the least-squares estimates and of and , respectively, and explain why in this case they are the same as the maximum-likelihood estimates.
(ii) Determine the maximum-likelihood estimate of and find a multiple of it which is an unbiased estimate of .
(iii) Determine the joint distribution of and .
(iv) Explain carefully how you would test the hypothesis against the alternative .
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1.I.8C
Part IB, 2007 commentState and prove the max-flow min-cut theorem for network flows.
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2.I.9C
Part IB, 2007 commentConsider the game with payoff matrix
where the entry is the payoff to the row player if the row player chooses row and the column player chooses column .
Find the value of the game and the optimal strategies for each player.
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Part IB, 2007
commentState and prove the Lagrangian sufficiency theorem.
Solve the problem
where and are non-negative constants satisfying .
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4.II
Part IB, 2007 commentConsider the linear programming problem
(i) After adding slack variables and and performing one iteration of the simplex algorithm, the following tableau is obtained.
\begin{tabular}{c|rrrrrr|c} & & & & & & & \ \hline & & 1 & 2 & & 0 & 0 & \ & 6 & 0 & & & 1 & 0 & 3 \ & 1 & 0 & & 2 & 0 & 1 & 15 \ \hline Payoff & & 0 & 4 & & 0 & 0 & \end{tabular}
Complete the solution of the problem.
(ii) Now suppose that the problem is amended so that the objective function becomes
Find the solution of this new problem.
(iii) Formulate the dual of the problem in (ii) and identify the optimal solution to the dual.
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1.II.19C
Part IB, 2007 commentConsider a Markov chain on states with transition matrix , where , so that 0 and are absorbing states. Let
be the event that the chain is absorbed in 0 . Assume that for .
Show carefully that, conditional on the event is a Markov chain and determine its transition matrix.
Now consider the case where , for . Suppose that , and that the event occurs; calculate the expected number of transitions until the chain is first in the state 0 .
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2.II.20C
Part IB, 2007 commentConsider a Markov chain with state space and transition matrix given by
and otherwise, where .
For each value of , determine whether the chain is transient, null recurrent or positive recurrent, and in the last case find the invariant distribution.
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3.I.9C
Part IB, 2007 commentConsider a Markov chain with state space and transition matrix
where and .
Calculate for each .
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4.I.9C
Part IB, 2007 commentFor a Markov chain with state space , define what is meant by the following:
(i) states communicate;
(ii) state is recurrent.
Prove that communication is an equivalence relation on and that if two states communicate and is recurrent then is recurrent.
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1.I.1F
Part II, 2007 commentState the prime number theorem, and Bertrand's postulate.
Let be a finite set of prime numbers, and write for the number of positive integers no larger than , all of whose prime factors belong to . Prove that
where denotes the number of elements in . Deduce that, if is a strictly positive integer, we have
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2.I.1F
Part II, 2007 commentLet be an odd prime number. Prove that 2 is a quadratic residue modulo when . Deduce that, if is a prime number strictly greater than 3 with such that is also a prime number, then is necessarily composite. Why does the argument break down for ?
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Part II, 2007
commentDetermine the continued fraction of . Deduce two pairs of solutions in positive integers of the equation
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3.II.11F
Part II, 2007 commentState the Chinese remainder theorem. Let be an odd positive integer. If is divisible by the square of a prime number , prove that there exists an integer such that but .
Define the Jacobi symbol
for any non-zero integer . Give a numerical example to show that
does not imply in general that is a square modulo . State and prove the law of quadratic reciprocity for the Jacobi symbol.
[You may assume the law of quadratic reciprocity for the Legendre symbol.]
Assume now that is divisible by the square of a prime number. Prove that there exists an integer with such that the congruence
does not hold. Show further that this congruence fails to hold for at least half of all relatively prime residue classes modulo .
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4.I.1F
Part II, 2007 commentProve Legendre's formula relating and for any positive real number . Use this formula to compute .
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4.II.11F
Part II, 2007 commentLet be a prime number, and let be a polynomial with integer coefficients, whose leading coefficient is not divisible by . Prove that the congruence
has at most solutions, where is the degree of .
Deduce that all coefficients of the polynomial
must be divisible by , and prove that:
(i) ;
(ii) if is odd, the numerator of the fraction
is divisible by .
Assume now that . Show by example that (i) cannot be strengthened to .
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1.I.2F
Part II, 2007 commentLet be an integer with . Are the following statements true or false? Give proofs.
(i) There exists a real polynomial of degree such that
for all real .
(ii) There exists a real polynomial of degree such that
for all real .
(iii) There exists a real polynomial of degree such that
for all real .
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2.II.12F
Part II, 2007 comment(i) Suppose that is continuous. Prove the theorem of Bernstein which states that, if we write
for , then uniformly as
(ii) Let and let be distinct points in . We write
for every continuous function . Show that, if
for all polynomials of degree or less, then for all and
(iii) If satisfies the conditions set out in (ii), show that
as whenever is continuous.
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Part II, 2007
commentWrite
Suppose that is a convex, compact subset of with . Show that there is a unique point such that
for all .
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3.II.12F
Part II, 2007 comment(i) State and prove Liouville's theorem on approximation of algebraic numbers by rationals.
(ii) Consider the continued fraction
where the are strictly positive integers. You may assume the following algebraic facts about the th convergent .
Show that
Give explicit values for so that is transcendental and prove that you have done SO.
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Part II, 2007
commentState a version of Runge's theorem and use it to prove the following theorem:
Let and define by the condition
for all and all . (We take to be the positive square root.) Then there exists a sequence of analytic functions such that for each as .
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4.I.2F
Part II, 2007 commentState Brouwer's fixed point theorem for a triangle in two dimensions.
Let be a matrix with real positive entries and such that all its columns are non-zero vectors. Show that has an eigenvector with positive entries.
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Part II, 2007
commentShow that there are two ways to embed a regular tetrahedron in a cube so that the vertices of the tetrahedron are also vertices of . Show that the symmetry group of permutes these tetrahedra and deduce that the symmetry group of is isomorphic to the Cartesian product of the symmetric group and the cyclic group .
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1.II.12G
Part II, 2007 commentDefine the Hausdorff -dimensional measure and the Hausdorff dimension of a subset of .
Set . Define the Cantor set and show that its Hausdorff -dimensional measure is at most
Let be independent Bernoulli random variables that take the values 0 and 2 , each with probability . Define
Show that is a random variable that takes values in the Cantor set .
Let be a subset of with . Show that and deduce that, for any set , we have
Hence, or otherwise, prove that and that the Cantor set has Hausdorff dimension .
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2.I.3G
Part II, 2007 commentExplain what is meant by a lattice in the Euclidean plane . Prove that such a lattice is either for some vector or else for two linearly independent vectors in .
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3.I.3G
Part II, 2007 commentLet be a 2-dimensional Euclidean crystallographic group. Define the lattice and point group corresponding to .
Prove that any non-trivial rotation in the point group of must have order or
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4.I
Part II, 2007 commentLet be a circle on the Riemann sphere. Explain what it means to say that two points of the sphere are inverse points for the circle . Show that, for each point on the Riemann sphere, there is a unique point with inverse points. Define inversion in .
Prove that the composition of an even number of inversions is a Möbius transformation.
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4.II.12G
Part II, 2007 commentExplain what it means to say that a group is a Kleinian group. What is the definition of the limit set for the group ? Prove that a fixed point of a parabolic element in must lie in the limit set.
Show that the matrix represents a parabolic transformation for any non-zero choice of the complex numbers and . Find its fixed point.
The Gaussian integers are . Let be the set of Möbius transformations with and . Prove that is a Kleinian group. For each point with non-zero integers, find a parabolic transformation that fixes . Deduce that the limit set for is all of the Riemann sphere.
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1.I.4G
Part II, 2007 commentLet and be alphabets of sizes and . What does it mean to say that is a decipherable code? State the inequalities of Kraft and Gibbs, and deduce that if letters are drawn from with probabilities then the expected word length is at least .
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2.I.4G
Part II, 2007 commentBriefly explain how and why a signature scheme is used. Describe the El Gamal scheme.
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1.II.11G
Part II, 2007 commentDefine the bar product of linear codes and , where is a subcode of . Relate the rank and minimum distance of to those of and . Show that if denotes the dual code of , then
Using the bar product construction, or otherwise, define the Reed-Muller code for . Show that if , then the dual of is again a Reed-Muller code.
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Part II, 2007
commentCompute the rank and minimum distance of the cyclic code with generator polynomial and parity-check polynomial . Now let be a root of in the field with 8 elements. We receive the word . Verify that , and hence decode using minimum-distance decoding.
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2.II.11G
Part II, 2007 commentDefine the capacity of a discrete memoryless channel. State Shannon's second coding theorem and use it to show that the discrete memoryless channel with channel matrix
has capacity .
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4.I.4G
Part II, 2007 commentWhat is a linear feedback shift register? Explain the Berlekamp-Massey method for recovering the feedback polynomial of a linear feedback shift register from its output. Illustrate in the case when we observe output
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1.I.5I
Part II, 2007 commentAccording to the Independent newspaper (London, 8 March 1994) the Metropolitan Police in London reported 30475 people as missing in the year ending March 1993. For those aged 18 or less, 96 of 10527 missing males and 146 of 11363 missing females were still missing a year later. For those aged 19 and above, the values were 157 of 5065 males and 159 of 3520 females. This data is summarised in the table below.
\begin{array}{rrrrr} & \multicolumn{3}{r}{\text { age }} \\ 1 & \text { Kender } & \text { M } & 96 & 10527 \\ 2 & \text { Kid } & \text { F } & 146 & 11363 \\ 3 & \text { Adult } & \text { M } & 157 & 5065 \\ 4 & \text { Adult } & \text { F } & 159 & 3520 \end{array}
Explain and interpret the commands and (slightly abbreviated) output below. You should describe the model being fitted, explain how the standard errors are calculated, and comment on the hypothesis tests being described in the summary. In particular, what is the worst of the four categories for the probability of remaining missing a year later?

For a person who was missing in the year ending in March 1993, find a formula, as a function of age and gender, for the estimated expected probability that they are still missing a year later.
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1.II.13I
Part II, 2007 commentThis problem deals with data collected as the number of each of two different strains of Ceriodaphnia organisms are counted in a controlled environment in which reproduction is occurring among the organisms. The experimenter places into the containers a varying concentration of a particular component of jet fuel that impairs reproduction. Hence it is anticipated that as the concentration of jet fuel grows, the mean number of organisms should decrease.
The table below gives a subset of the data. The full dataset has rows. The first column provides the number of organisms, the second the concentration of jet fuel (in grams per litre) and the third specifies the strain of the organism.
Explain and interpret the commands and (slightly abbreviated) output below. In particular, you should describe the model being fitted, explain how the standard errors are calculated, and comment on the hypothesis tests being described in the summary.

The following code fits two very similar models. Briefly explain the difference between these models and the one above. Motivate the fitting of these models in light of
Part II 2007 the summary from the fit of the one above.

Denote by the three hypotheses being fitted in sequence above.
Explain the hypothesis tests, including an approximate test of the fit of , that can be performed using the output from the following code. Use these numbers to comment on the most appropriate model for the data.
, fit2$dev, fit3$dev)
[1]
[1]
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2.I.5I
Part II, 2007 commentConsider the linear regression setting where the responses are assumed independent with means . Here is a vector of known explanatory variables and is a vector of unknown regression coefficients.
Show that if the response distribution is Laplace, i.e.,
then the maximum likelihood estimate of is obtained by minimising
Obtain the maximum likelihood estimate for in terms of .
Briefly comment on why the Laplace distribution cannot be written in exponential dispersion family form.
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3.I.5I
Part II, 2007 commentConsider two possible experiments giving rise to observed data where
- The data are realizations of independent Poisson random variables, i.e.,
where , with an unknown (possibly vector) parameter. Write for the maximum likelihood estimator (m.l.e.) of and for the th fitted value under this model.
- The data are components of a realization of a multinomial random 'vector'
where the are non-negative integers with
Write for the m.l.e. of and for the th fitted value under this model.
Show that, if
then and for all . Explain the relevance of this result in the context of fitting multinomial models within a generalized linear model framework.
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4.I.5I
Part II, 2007 commentConsider the normal linear model in vector notation, where
i.i.d. ,
where is known and is of full rank . Give expressions for maximum likelihood estimators and of and respectively, and state their joint distribution.
Suppose that there is a new pair , independent of , satisfying the relationship
We suppose that is known, and estimate by . State the distribution of
Find the form of a -level prediction interval for .
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4.II.13I
Part II, 2007 commentLet have a Gamma distribution with density
Show that the Gamma distribution is of exponential dispersion family form. Deduce directly the corresponding expressions for and in terms of and . What is the canonical link function?
Let . Consider a generalised linear model (g.l.m.) for responses with random component defined by the Gamma distribution with canonical link , so that , where is the vector of unknown regression coefficients and is the vector of known values of the explanatory variables for the th observation, .
Obtain expressions for the score function and Fisher information matrix and explain how these can be used in order to approximate , the maximum likelihood estimator (m.l.e.) of .
[Use the canonical link function and assume that the dispersion parameter is known.]
Finally, obtain an expression for the deviance for a comparison of the full (saturated) model to the g.l.m. with canonical link using the m.l.e. (or estimated mean .
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1.I.6B
Part II, 2007 commentA chemostat is a well-mixed tank of given volume that contains water in which lives a population of bacteria that consume nutrient whose concentration is per unit volume. An inflow pipe supplies a solution of nutrient at concentration and at a constant flow rate of units of volume per unit time. The mixture flows out at the same rate through an outflow pipe. The bacteria consume the nutrient at a rate , where
and the bacterial population grows at a rate , where .
Write down the differential equations for and show that they can be rescaled into the following form:
where are positive constants, to be found.
Show that this system of equations has a non-trivial steady state if and , and that it is stable.
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Part II, 2007
commentA field contains seed-producing poppies in August of year . On average each poppy produces seeds, a number that is assumed not to vary from year to year. A fraction of seeds survive the winter and a fraction of those germinate in May of year . A fraction of those that survive the next winter germinate in year . Show that satisfies the following difference equation:
Write down the general solution of this equation, and show that the poppies in the field will eventually die out if
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2.II.13B
Part II, 2007 commentShow that the concentration of a diffusible chemical substance in a stationary medium satisfies the partial differential equation
where is the diffusivity and is the rate of supply of the chemical.
A finite amount of the chemical, , is supplied at the origin at time , and spreads out in a spherically symmetric manner, so that for , where is the radial coordinate. The diffusivity is given by , for constant . Show, by dimensional analysis or otherwise, that it is appropriate to seek a similarity solution in which
where are constants to be determined, and derive the ordinary differential equation satisfied by .
Solve this ordinary differential equation, subject to appropriate boundary conditions, and deduce that the chemical occupies a finite spherical region of radius
[Note: in spherical polar coordinates
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3.I.6B
Part II, 2007 commentConsider a birth and death process in which births always give rise to two offspring, with rate , while the death rate per individual is .
Write down the master equation (or probability balance equation) for this system.
Show that the population mean is given by
where is the initial population mean, and that the population variance satisfies
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3.II.13B
Part II, 2007 commentThe number density of a population of cells is . The cells produce a chemical whose concentration is and respond to it chemotactically. The equations governing and are
(i) Give a biological interpretation of each term in these equations, where you may assume that and are all positive.
(ii) Show that there is a steady-state solution that is stable to spatially invariant disturbances.
(iii) Analyse small, spatially-varying perturbations to the steady state that satisfy for any variable , and show that a chemotactic instability is possible if
(iv) Find the critical value of at which the instability first appears as is increased.
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4.I.6B
Part II, 2007 commentThe non-dimensional equations for two competing populations are
Explain the meaning of each term in these equations.
Find all the fixed points of this system when and , and investigate their stability.
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1.I.7E
Part II, 2007 commentGiven a non-autonomous th-order differential equation
with , explain how it may be written in the autonomous first-order form for suitably chosen vectors and .
Given an autonomous system in , define the corresponding flow . What is equal to? Define the orbit through and the limit set of . Define a homoclinic orbit.
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3.II.14E
Part II, 2007 commentThe Lorenz equations are
where and are positive constants and .
(i) Show that the origin is globally asymptotically stable for by considering a function with a suitable choice of constants and
(ii) State, without proof, the Centre Manifold Theorem.
Show that the fixed point at the origin is nonhyperbolic at . What are the dimensions of the linear stable and (non-extended) centre subspaces at this point?
(iii) Let from now on. Make the substitutions and and derive the resulting equations for and .
The extended centre manifold is given by
where and can be expanded as power series about . What is known about and from the Centre Manifold Theorem? Assuming that , determine correct to and to . Hence obtain the evolution equation on the extended centre manifold correct to , and identify the type of bifurcation.
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2.I.7E
Part II, 2007 commentFind and classify the fixed points of the system
What are the values of their Poincaré indices? Prove that there are no periodic orbits. Sketch the phase plane.
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4.II.14E
Part II, 2007 commentConsider the one-dimensional map defined by
where and are constants, is a parameter and .
(i) Find the fixed points of and determine the linear stability of . Hence show that there are bifurcations at , at and, if , at .
Sketch the bifurcation diagram for each of the cases:
In each case show the locus and stability of the fixed points in the -plane, and state the type of each bifurcation. [Assume that there are no further bifurcations in the region sketched.]
(ii) For the case (i.e. , you may assume that
Show that there are at most three 2-cycles and determine when they exist. By considering , or otherwise, show further that one 2-cycle is always unstable when it exists and that the others are unstable when . Sketch the bifurcation diagram showing the locus and stability of the fixed points and 2 -cycles. State briefly what you would expect to occur in the region .
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3.I.7E
Part II, 2007 commentState the Poincaré-Bendixson Theorem for a system in .
Prove that if then the system
has a periodic orbit in the region .
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4.I.7E
Part II, 2007 commentBy considering the binary representation of the sawtooth for , show that:
(i) has sensitive dependence on initial conditions on .
(ii) has topological transitivity on .
(iii) Periodic points are dense in .
Find all the 4-cycles of and express them as fractions.
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1.I.8B
Part II, 2007 commentThe coefficients and of the differential equation
are analytic in the punctured disc , and and are linearly independent solutions in the neighbourhood of the point in the disc. By considering the effect of analytically continuing and , show that the equation has a non-trivial solution of the form
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2.I.8B
Part II, 2007 commentThe function is defined by
For what values of is analytic?
By considering , where is the Riemann zeta function which you may assume is given by
show that . Deduce from this result that the analytic continuation of is an entire function. [You may use properties of without proof.]
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3.I.8B
Part II, 2007 commentLet and be any two linearly independent branches of the -function
where , and let be the Wronskian of and .
(i) How is related to the Wronskian of the principal branches of the -function at ?
(ii) Show that is an entire function.
(iii) Given that is bounded as , show that
where is a non-zero constant.
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1.II.14B
Part II, 2007 commentThe function is defined by
where is a constant (which is not an integer). The path of integration, , is the Pochhammer contour, defined as follows. It starts at a point on the axis between 0 and 1 , then it circles the points 1 and 0 in the negative sense, then it circles the points 1 and 0 in the positive sense, returning to . At the start of the path, and the integrand is defined at other points on by analytic continuation along .
(i) For what values of is analytic? Give brief reasons for your answer.
(ii) Show that, in the case and ,
where is the Beta function.
(iii) Deduce that the only singularities of are simple poles. Explain carefully what happens if is a positive integer.
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4.I.8B
Part II, 2007 commentThe hypergeometric function is defined by
where and is a constant determined by the condition .
(i) Express in terms of Gamma functions.
(ii) By considering the th derivative , show that .
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2.II.14B
Part II, 2007 commentShow that the equation
has solutions of the form , where
provided that is suitably chosen.
Hence find the general solution, evaluating the integrals explicitly. Show that the general solution is entire, but that there is no solution that satisfies and .
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1.I.9C
Part II, 2007 commentThe action for a system with generalized coordinates, , for a time interval is given by
where is the Lagrangian, and where the end point values and are fixed at specified values. Derive Lagrange's equations from the principle of least action by considering the variation of for all possible paths.
What is meant by the statement that a particular coordinate is ignorable? Show that there is an associated constant of the motion, to be specified in terms of .
A particle of mass is constrained to move on the surface of a sphere of radius under a potential, , for which the Lagrangian is given by
Identify an ignorable coordinate and find the associated constant of the motion, expressing it as a function of the generalized coordinates. Evaluate the quantity
in terms of the same generalized coordinates, for this case. Is also a constant of the motion? If so, why?
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2.II.15C
Part II, 2007 comment(a) A Hamiltonian system with degrees of freedom is described by the phase space coordinates and momenta . Show that the phase-space volume element
is conserved under time evolution.
(b) The Hamiltonian, , for the system in part (a) is independent of time. Show that if is a constant of the motion, then the Poisson bracket vanishes. Evaluate when
and
where the potential depends on the only through quantities of the form for .
(c) For a system with one degree of freedom, state what is meant by the transformation
being canonical. Show that the transformation is canonical if and only if the Poisson bracket .
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2.I.9C
Part II, 2007 commentThe Lagrangian for a particle of mass and charge moving in a magnetic field with position vector is given by
where the vector potential , which does not depend on time explicitly, is related to the magnetic field through
Write down Lagrange's equations and use them to show that the equation of motion of the particle can be written in the form
Deduce that the kinetic energy, , is constant.
When the magnetic field is of the form for some specified function , show further that
where and are constants.
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3.I.9C
Part II, 2007 commentA particle of mass is constrained to move in the horizontal plane, around a circle of fixed radius whose centre is at the origin of a Cartesian coordinate system . A second particle of mass is constrained to move around a circle of fixed radius that also lies in a horizontal plane, but whose centre is at . It is given that the Lagrangian of the system can be written as
using the particles' cylindrical polar angles and as generalized coordinates. Deduce the equations of motion and use them to show that is constant, and that obeys an equation of the form
where is a constant to be determined.
Find two values of corresponding to equilibria, and show that one of the two equilibria is stable. Find the period of small oscillations about the stable equilibrium.
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4.II.15C
Part II, 2007 commentThe Hamiltonian for an oscillating particle with one degree of freedom is
The mass is a constant, and is a function of time alone. Write down Hamilton's equations and use them to show that
Now consider a case in which is constant and the oscillation is exactly periodic. Denote the constant value of in that case by . Consider the quantity , where the integral is taken over a single oscillation cycle. For any given function show that can be expressed as a function of and alone, namely
where the sign of the integrand alternates between the two halves of the oscillation cycle. Let be the period of oscillation. Show that the function has partial derivatives
You may assume without proof that and may be taken inside the integral.
Now let change very slowly with time , by a negligible amount during an oscillation cycle. Assuming that, to sufficient approximation,
where is the average value of over an oscillation cycle, and that
deduce that , carefully explaining your reasoning.
When
with a positive integer and positive, deduce that
for slowly-varying , where is a constant.
[Do not try to solve Hamilton's equations. Rather, consider the form taken by . ]
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4.I
Part II, 2007 comment(a) Show that the principal moments of inertia for the oblate spheroid of mass defined by
are given by . Here is the semi-major axis and is the eccentricity.
[You may assume that a sphere of radius a has principal moments of inertia .]
(b) The spheroid in part (a) rotates about an axis that is not a principal axis. Euler's equations governing the angular velocity as viewed in the body frame are
and
Show that is constant. Show further that the angular momentum vector precesses around the axis with period
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1.I.10A
Part II, 2007 commentDescribe the motion of light rays in an expanding universe with scale factor , and derive the redshift formula
where the light is emitted at time and observed at time .
A galaxy at comoving position is observed to have a redshift . Given that the galaxy emits an amount of energy per unit time, show that the total energy per unit time crossing a sphere centred on the galaxy and intercepting the earth is . Hence, show that the energy per unit time per unit area passing the earth is
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2.I.10A
Part II, 2007 commentThe number density of photons in thermal equilibrium at temperature takes the form
At time and temperature , photons decouple from thermal equilibrium. By considering how the photon frequency redshifts as the universe expands, show that the form of the equilibrium frequency distribution is preserved, with the temperature for defined by
Show that the photon number density and energy density can be expressed in the form
where the constants and need not be evaluated explicitly.
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1.II.15A
Part II, 2007 commentIn a homogeneous and isotropic universe, the scale factor obeys the Friedmann equation
where is the matter density, which, together with the pressure , satisfies
Here, is a constant curvature parameter. Use these equations to show that the rate of change of the Hubble parameter satisfies
Suppose that an expanding Friedmann universe is filled with radiation (density and pressure as well as a "dark energy" component (density and pressure . Given that the energy densities of these two components are measured today to be
show that the curvature parameter must satisfy . Hence derive the following relations for the Hubble parameter and its time derivative:
Show qualitatively that universes with will recollapse to a Big Crunch in the future. [Hint: Sketch and versus for representative values of .]
For , find an explicit solution for the scale factor satisfying . Find the limiting behaviours of this solution for large and small . Comment briefly on their significance.
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3.I.10A
Part II, 2007 commentThe number density of a non-relativistic species in thermal equilibrium is given by
Suppose that thermal and chemical equilibrium is maintained between protons p (mass , degeneracy ), neutrons (mass , degeneracy ) and helium-4 nuclei mass , degeneracy ) via the interaction
where you may assume the photons have zero chemical potential . Given that the binding energy of helium-4 obeys , show that the ratio of the number densities can be written as
Explain briefly why the baryon-to-photon ratio remains constant during the expansion of the universe, where and .
By considering the fractional densities of the species , re-express the ratio ( ) in the form
Given that , verify (very approximately) that this ratio approaches unity when . In reality, helium-4 is not formed until after deuterium production at a considerably lower temperature. Explain briefly the reason for this delay.
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4.I.10A
Part II, 2007 commentThe equation governing density perturbation modes in a matter-dominated universe (with ) is
where is the comoving wavevector. Find the general solution for the perturbation, showing that there is a growing mode such that
Show that the physical wavelength corresponding to the comoving wavenumber crosses the Hubble radius at a time given by
According to inflationary theory, the amplitude of the variance at horizon-crossing is constant, that is, where and (the volume) are constants. Given this amplitude and the results obtained above, deduce that the power spectrum today takes the form
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3.II.15A
Part II, 2007 commentA spherically symmetric star with outer radius has mass density and pressure , where is the distance from the centre of the star. Show that hydrostatic equilibrium implies the pressure support equation,
where is the mass inside radius . State without proof any results you may need.
Write down an integral expression for the total gravitational potential energy of the star. Hence use to deduce the virial theorem
where is the average pressure and is the volume of the star.
Given that a non-relativistic ideal gas obeys and that an ultrarelativistic gas obeys , where is the kinetic energy, discuss briefly the gravitational stability of a star in these two limits.
At zero temperature, the number density of particles obeying the Pauli exclusion principle is given by
where is the Fermi momentum, is the degeneracy and is Planck's constant. Deduce that the non-relativistic internal energy of these particles is
where is the mass of a particle. Hence show that the non-relativistic Fermi degeneracy pressure satisfies
Use the virial theorem to estimate that the radius of a star supported by Fermi degeneracy pressure is approximately
where is the total mass of the star.
[Hint: Assume and note that
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1.II.16G
Part II, 2007 commentBy a directed set in a poset , we mean a nonempty subset such that any pair of elements of has an upper bound in . We say is directed-complete if each directed subset has a least upper bound in . Show that a poset is complete if and only if it is directed-complete and has joins for all its finite subsets. Show also that, for any two sets and , the set of partial functions from to , ordered by extension, is directed-complete.
Let be a directed-complete poset, and an order-preserving map which is inflationary, i.e. satisfies for all . We define a subset to be closed if it satisfies , and is also closed under joins of directed sets (i.e., and directed imply ). We write to mean that every closed set containing also contains . Show that is a partial order on , and that implies . Now consider the set of all functions which are order-preserving and satisfy for all . Show that is closed under composition of functions, and deduce that, for each , the set is directed. Defining for each , show that the function belongs to , and deduce that is the least fixed point of lying above , for each .
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2.II.16G
Part II, 2007 commentExplain carefully what is meant by a deduction in the propositional calculus. State the completeness theorem for the propositional calculus, and deduce the compactness theorem.
Let be three pairwise-disjoint sets of primitive propositions, and suppose given compound propositions and such that holds. Let denote the set
If is any valuation making all the propositions in true, show that the set
is consistent. Deduce that is inconsistent, and hence show that there exists such that and both hold.
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3.II.16G
Part II, 2007 commentWrite down the recursive definitions of ordinal addition, multiplication and exponentiation. Prove carefully that for all , and hence show that for each non-zero ordinal there exists a unique such that
Deduce that any non-zero ordinal has a unique representation of the form
where and are non-zero natural numbers.
Two ordinals are said to be commensurable if we have neither nor . Show that and are commensurable if and only if there exists such that both and lie in the set
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4.II.16G
Part II, 2007 commentExplain what is meant by a well-founded binary relation on a set.
Given a set , we say that a mapping is recursive if, given any set equipped with a mapping , there exists a unique such that , where denotes the mapping . Show that is recursive if and only if the relation is well-founded.
[If you need to use any form of the recursion theorem, you should prove it.]
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1.II.17H
Part II, 2007 commentLet be a connected cubic graph drawn in the plane with each edge in the boundary of two distinct faces. Show that the associated map is 4 -colourable if and only if is 3 -edge colourable.
Is the above statement true if the plane is replaced by the torus and all faces are required to be simply connected? Give a proof or a counterexample.
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2.II.17H
Part II, 2007 commentThe Ramsey number of a graph is the smallest such that in any red/blue colouring of the edges of there is a monochromatic copy of .
Show that for every .
Let be the graph on four vertices obtained by adding an edge to a triangle. Show that .
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3.II.17H
Part II, 2007 commentLet be a bipartite graph with vertex classes and , each of size . State and prove Hall's theorem giving a necessary and sufficient condition for to contain a perfect matching.
A vertex is flexible if every edge from is contained in a perfect matching. Show that if for every subset of with , then every is flexible.
Show that whenever contains a perfect matching, there is at least one flexible .
Give an example of such a where no of minimal degree is flexible.
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4.II.17H
Part II, 2007 commentLet be a graph with vertices and edges. Show that if contains no , then .
Let denote the number of subgraphs of isomorphic to . Show that if , then contains at least paths of length 2 . By considering the numbers of vertices joined to each pair of vertices of , deduce that
Now let be the random graph on in which each pair of vertices is joined independently with probability . Find the expectation of . Deduce that if , then
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1.II.18F
Part II, 2007 commentLet be field extensions. Define the degree of the field extension , and state and prove the tower law.
Now let be a finite field. Show , for some prime and positive integer . Show also that contains a subfield of order if and only if .
If is an irreducible polynomial of degree over the finite field , determine its Galois group.
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2.II.18F
Part II, 2007 commentLet , where is a primitive th root of unity and . Prove that there is an injective group homomorphism .
Show that, if is an intermediate subfield of , then is Galois. State carefully any results that you use.
Give an example where is non-trivial but is not surjective. Show that is surjective when and is a prime.
Determine all the intermediate subfields of and the automorphism groups . Write the quadratic subfield in the form for some .
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3.II.18F
Part II, 2007 comment(i) Let be the splitting field of the polynomial over . Describe the field , the Galois group , and the action of on .
(ii) Let be the splitting field of the polynomial over . Describe the field and determine .
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4.II.18F
Part II, 2007 commentLet be a monic polynomial, a splitting field for the roots of in . Let be the discriminant of . Explain why is a polynomial function in the coefficients of , and determine when .
Compute the Galois group of the polynomial .
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1.II.19H
Part II, 2007 commentA finite group has seven conjugacy classes and the values of five of its irreducible characters are given in the following table.
Calculate the number of elements in the various conjugacy classes and complete the character table.
[You may not identify with any known group, unless you justify doing so.]
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2.II.19H
Part II, 2007 commentLet be a finite group and let be its centre. Show that if is a complex irreducible representation of , assumed to be faithful (that is, the kernel of is trivial), then is cyclic.
Now assume that is a p-group (that is, the order of is a power of the prime , and assume that is cyclic. If is a faithful representation of , show that some irreducible component of is faithful.
[You may use without proof the fact that, since is a p-group, is non-trivial and any non-trivial normal subgroup of intersects non-trivially.]
Deduce that a finite -group has a faithful irreducible representation if and only if its centre is cyclic.
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3.II.19H
Part II, 2007 commentLet be a finite group with a permutation action on the set . Describe the corresponding permutation character . Show that the multiplicity in of the principal character equals the number of orbits of on .
Assume that is transitive on , with . Show that contains an element which is fixed-point-free on , that is, for all in .
Assume that , with an irreducible character of , for some natural number . Show that .
[You may use without proof any facts about algebraic integers, provided you state them correctly.]
Explain how the action of on induces an action of on . Assume that has orbits on . If now
with distinct irreducible characters of , and natural numbers, show that . Deduce that, if , then and .
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4.II.19H
Part II, 2007 commentWrite an essay on the representation theory of .
Your answer should include a description of each irreducible representation and an explanation of how to decompose arbitrary representations into a direct sum of these.
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1.II.20H
Part II, 2007 commentLet .
(a) Show that and that the discriminant is equal to .
(b) Show that 2 ramifies in by showing that , and that is not a principal ideal. Show further that with . Deduce that neither nor is a principal ideal, but .
(c) Show that 5 splits in by showing that , and that
Deduce that has trivial class in the ideal class group of . Conclude that the ideal class group of is cyclic of order six.
[You may use the fact that
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2.II.20H
Part II, 2007 commentLet and put .
(a) Show that 2,3 and are irreducible elements in . Deduce from the equation
that is not a principal ideal domain.
(b) Put and . Show that
Deduce that has class number 2 .
(c) Show that is the fundamental unit of . Hence prove that all solutions in integers of the equation are given by
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4.II.20H
Part II, 2007 commentLet be a finite extension of and let be its ring of integers. We will assume that for some . The minimal polynomial of will be denoted by . For a prime number let
be the decomposition of into distinct irreducible monic factors . Let be a polynomial whose reduction modulo is . Show that
are the prime ideals of containing , that these are pairwise different, and
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1.II.21H
Part II, 2007 comment(i) Compute the fundamental group of the Klein bottle. Show that this group is not abelian, for example by defining a suitable homomorphism to the symmetric group .
(ii) Let be the closed orientable surface of genus 2 . How many (connected) double coverings does have? Show that the fundamental group of admits a homomorphism onto the free group on 2 generators.
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2.II.21H
Part II, 2007 commentState the Mayer-Vietoris sequence for a simplicial complex which is a union of two subcomplexes and . Define the homomorphisms in the sequence (but do not check that they are well-defined). Prove exactness of the sequence at the term .
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Part II, 2007
commentDefine what it means for a group to act on a topological space . Prove that, if acts freely, in a sense that you should specify, then the quotient map is a covering map and there is a surjective group homomorphism from the fundamental group of to .
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4.II
Part II, 2007 commentCompute the homology of the space obtained from the torus by identifying to a point and to a point, for two distinct points and in
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1.II.22G
Part II, 2007 commentLet be a normed vector space over . Define the dual space and show directly that is a Banach space. Show that the map defined by , for all , is a linear map. Using the Hahn-Banach theorem, show that is injective and .
Give an example of a Banach space for which is not surjective. Justify your answer.
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2.II.22G
Part II, 2007 commentLet be a Banach space, a normed vector space, and a bounded linear map. Assume that is of second category in . Show that is surjective and is open whenever is open. Show that, if is also injective, then exists and is bounded.
Give an example of a continuous map such that is of second category in but is not surjective. Give an example of a continuous surjective map which does not take open sets to open sets.
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3.II.21G
Part II, 2007 commentState and prove the Arzela-Ascoli theorem.
Let be a positive integer. Consider the subset consisting of all thrice differentiable solutions to the differential equation
with
Show that this set is totally bounded as a subset of .
[It may be helpful to consider interior maxima.]
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4.II.22G
Part II, 2007 commentLet be a Banach space and a bounded linear map. Define the spectrum , point spectrum , resolvent , and resolvent set . Show that the spectrum is a closed and bounded subset of . Is the point spectrum always closed? Justify your answer.
Now suppose is a Hilbert space, and is self-adjoint. Show that the point spectrum is real.
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1.II.23F
Part II, 2007 commentDefine a complex structure on the unit sphere using stereographic projection charts . Let be an open set. Show that a continuous non-constant map is holomorphic if and only if is a meromorphic function. Deduce that a non-constant rational function determines a holomorphic map . Define what is meant by a rational function taking the value with multiplicity at infinity.
Define the degree of a rational function. Show that any rational function satisfies and give examples to show that the bounds are attained. Is it true that the product satisfies , for any non-constant rational functions and ? Justify your answer.
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2.II.23F
Part II, 2007 commentA function is defined for by
where is a complex parameter with . Prove that this series converges uniformly on the subsets for and deduce that is holomorphic on .
You may assume without proof that
for all . Let be the logarithmic derivative . Show that
for all . Deduce that has only one zero in the parallelogram with vertices . Find all of the zeros of
Let be the lattice in generated by 1 and . Show that, for , the formula
gives a -periodic meromorphic function if and only if . Deduce that is -periodic.
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3.II.22F
Part II, 2007 comment(i) Let and be compact connected Riemann surfaces and a non-constant holomorphic map. Define the branching order at showing that it is well defined. Prove that the set of ramification points is finite. State the Riemann-Hurwitz formula.
Now suppose that and have the same genus . Prove that, if , then is biholomorphic. In the case when , write down an example where is not biholomorphic.
[The inverse mapping theorem for holomorphic functions on domains in may be assumed without proof if accurately stated.]
(ii) Let be a non-singular algebraic curve in . Describe, without detailed proofs, a family of charts for , so that the restrictions to of the first and second projections are holomorphic maps. Show that the algebraic curve
is non-singular. Find all the ramification points of the .
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4.II.23F
Part II, 2007 commentLet be a Riemann surface, a topological surface, and a continuous map. Suppose that every point admits a neighbourhood such that maps homeomorphically onto its image. Prove that has a complex structure such that is a holomorphic map.
A holomorphic map between Riemann surfaces is called a covering map if every has a neighbourhood with a disjoint union of open sets in , so that is biholomorphic for each . Suppose that a Riemann surface admits a holomorphic covering map from the unit . Prove that any holomorphic map is constant.
[You may assume any form of the monodromy theorem and basic results about the lifts of paths, provided that these are accurately stated.]
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1.II.24H
Part II, 2007 commentLet be a smooth map between manifolds without boundary. Recall that is a submersion if is surjective for all . The canonical submersion is the standard projection of onto for , given by
(i) Let be a submersion, and . Show that there exist local coordinates around and such that , in these coordinates, is the canonical submersion. [You may assume the inverse function theorem.]
(ii) Show that submersions map open sets to open sets.
(iii) If is compact and connected, show that every submersion is surjective. Are there submersions of compact manifolds into Euclidean spaces with ?
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2.II.24H
Part II, 2007 comment(i) What is a minimal surface? Explain why minimal surfaces always have non-positive Gaussian curvature.
(ii) A smooth map between two surfaces in 3-space is said to be conformal if
for all and all , where is a number which depends only on .
Let be a surface without umbilical points. Prove that is a minimal surface if and only if the Gauss map is conformal.
(iii) Show that isothermal coordinates exist around a non-planar point in a minimal surface.
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3.II.23H
Part II, 2007 comment(i) Let be a smooth map between manifolds without boundary. Define critical point, critical value and regular value. State Sard's theorem.
(ii) Explain how to define the degree modulo 2 of a smooth map , indicating clearly the hypotheses on and . Show that a smooth map with non-zero degree modulo 2 must be surjective.
(iii) Let be the torus of revolution obtained by rotating the circle in the -plane around the -axis. Describe the critical points and the critical values of the Gauss map of . Find the degree modulo 2 of . Justify your answer by means of a sketch or otherwise.
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4.II.24H
Part II, 2007 comment(i) What is a geodesic? Show that geodesics are critical points of the energy functional.
(ii) Let be a surface which admits a parametrization defined on an open subset of such that and , where is a function of alone and is a function of alone. Let be a geodesic and write . Show that
is independent of .
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1.II.25J
Part II, 2007 commentLet be a set and be a set system.
(a) Explain what is meant by a -system, a -system and a -algebra.
(b) Show that is a -algebra if and only if is a -system and a -system.
(c) Which of the following set systems are -systems, -systems or -algebras? Justify your answers. ( denotes the number of elements in .)
and is even ,
and is even or ,
and
(d) State and prove the theorem on the uniqueness of extension of a measure.
[You may use standard results from the lectures without proof, provided they are clearly stated.]
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2.II.25J
Part II, 2007 comment(a) State and prove the first Borel-Cantelli lemma. State the second Borel-Cantelli lemma.
(b) Let be a sequence of independent random variables that converges in probability to the limit . Show that is almost surely constant.
A sequence of random variables is said to be completely convergent to if
(c) Show that complete convergence implies almost sure convergence.
(d) Show that, for sequences of independent random variables, almost sure convergence also implies complete convergence.
(e) Find a sequence of (dependent) random variables that converges almost surely but does not converge completely.
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3.II.24J
Part II, 2007 commentLet be a finite measure space, i.e. , and let .
(a) Define the -norm of a measurable function , define the space and define convergence in
In the following you may use inequalities from the lectures without proof, provided they are clearly stated.
(b) Let . Show that in implies .
(c) Let be a bounded measurable function with . Let
Show that and .
By using Jensen's inequality, or otherwise, show that
Prove that
Observe that
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4.II.25J
Part II, 2007 commentLet be a measure space with and let be measurable.
(a) Define an invariant set and an invariant function .
What is meant by saying that is measure-preserving?
What is meant by saying that is ergodic?
(b) Which of the following functions to is ergodic? Justify your answer.
On the measure space with Lebesgue measure consider
On the discrete measure space consider
(c) State Birkhoff's almost everywhere ergodic theorem.
(d) Let be measure-preserving and let be bounded.
Prove that converges in for all .
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1.II.26J
Part II, 2007 commentAn open air rock concert is taking place in beautiful Pine Valley, and enthusiastic fans from the entire state of Alifornia are heading there long before the much anticipated event. The arriving cars have to be directed to one of three large (practically unlimited) parking lots, and situated near the valley entrance. The traffic cop at the entrance to the valley decides to direct every third car (in the order of their arrival) to a particular lot. Thus, cars and so on are directed to lot , cars to lot and cars to lot .
Suppose that the total arrival process , at the valley entrance is Poisson, of rate (the initial time is taken to be considerably ahead of the actual event). Consider the processes and where is the number of cars arrived in lot by time . Assume for simplicity that the time to reach a parking lot from the entrance is negligible so that the car enters its specified lot at the time it crosses the valley entrance.
(a) Give the probability density function of the time of the first arrival in each of the processes .
(b) Describe the distribution of the time between two subsequent arrivals in each of these processes. Are these times independent? Justify your answer.
(c) Which of these processes are delayed renewal processes (where the distribution of the first arrival time differs from that of the inter-arrival time)?
(d) What are the corresponding equilibrium renewal processes?
(e) Describe how the direction rule should be changed for and to become Poisson processes, of rate . Will these Poisson processes be independent? Justify your answer.
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2.II.26J
Part II, 2007 commentIn this question we work with a continuous-time Markov chain where the rate of jump may depend on but not on . A virus can be in one of strains , and it mutates to strain with rate from each strain . (Mutations are caused by the chemical environment.) Set .
(a) Write down the Q-matrix (the generator) of the chain in terms of and .
(b) If , that is, , what are the communicating classes of the chain ?
(c) From now on assume that . State and prove a necessary and sufficient condition, in terms of the numbers , for the chain to have a single communicating class (which therefore should be closed).
(d) In general, what is the number of closed communicating classes in the chain ? Describe all open communicating classes of .
(e) Find the equilibrium distribution of . Is the chain reversible? Justify your answer.
(f) Write down the transition matrix of the discrete-time jump chain for and identify its equilibrium distribution. Is the jump chain reversible? Justify your answer.
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3.II.25J
Part II, 2007 commentFor a discrete-time Markov chain, if the probability of transition does not depend on then the chain is reduced to a sequence of independent random variables (states). In this case, the chain forgets about its initial state and enters equilibrium after a single transition. In the continuous-time case, a Markov chain whose rates of transition depend on but not on still 'remembers' its initial state and reaches equilibrium only in the limit as the time grows indefinitely. This question is an illustration of this property.
A protean sea sponge may change its colour among varieties , under the influence of the environment. The rate of transition from colour to equals and does not depend on . Consider a Q-matrix with entries
where . Assume that and let be the continuous-time Markov chain with generator . Given , let be the matrix of transition probabilities in time in chain .
(a) State the exponential relation between the matrices and .
(b) Set . Check that are equilibrium probabilities for the chain . Is this a unique equilibrium distribution? What property of the vector with entries relative to the matrix is involved here?
(c) Let be a vector with components such that . Show that . Compute
(d) Now let denote the (column) vector whose entries are 0 except for the th one which equals 1. Observe that the th row of is . Prove that
(e) Deduce the expression for transition probabilities in terms of rates and their sum .
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4.II.26J
Part II, 2007 commentA population of rare Monarch butterflies functions as follows. At the times of a Poisson process of rate a caterpillar is produced from an egg. After an exponential time, the caterpillar is transformed into a pupa which, after an exponential time, becomes a butterfly. The butterfly lives for another exponential time and then dies. (The Poissonian assumption reflects the fact that butterflies lay a huge number of eggs most of which do not develop.) Suppose that all lifetimes are independent (of the arrival process and of each other) and let their rate be . Assume that the population is in an equilibrium and let be the number of caterpillars, the number of pupae and the number of butterflies (so that the total number of insects, in any metamorphic form, equals . Let be the equilibrium probability where
(a) Specify the rates of transitions for the resulting continuous-time Markov chain with states . (The rates are non-zero only when or and similarly for other co-ordinates.) Check that the holding rate for state is where .
(b) Let be the Q-matrix from (a). Consider the invariance equation . Verify that the only solution is
(c) Derive the marginal equilibrium probabilities and the conditional equilibrium probabilities .
(d) Determine whether the chain is positive recurrent, null-recurrent or transient.
(e) Verify that the equilibrium probabilities are the same as in the corresponding system (with the correct specification of the arrival rate and the service-time distribution).
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1.II.27I
Part II, 2007 commentSuppose that has density where . What does it mean to say that statistic is sufficient for ?
Suppose that , where is the parameter of interest, and is a nuisance parameter, and that the sufficient statistic has the form . What does it mean to say that the statistic is ancillary? If it is, how (according to the conditionality principle) do we test hypotheses on Assuming that the set of possible values for is discrete, show that is ancillary if and only if the density (probability mass function) factorises as
for some functions , and with the properties
for all , and .
Suppose now that are independent observations from a distribution, with density
Assuming that the criterion (*) holds also for observations which are not discrete, show that it is not possible to find sufficient for such that is ancillary when is regarded as a nuisance parameter, and is the parameter of interest.
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2.II.27I
Part II, 2007 comment(i) State Wilks' likelihood ratio test of the null hypothesis against the alternative , where . Explain when this test may be used.
(ii) Independent identically-distributed observations take values in the set , with common distribution which under the null hypothesis is of the form
for some , where is an open subset of some Euclidean space , . Under the alternative hypothesis, the probability mass function of the is unrestricted.
Assuming sufficient regularity conditions on to guarantee the existence and uniqueness of a maximum-likelihood estimator of for each , show that for large the Wilks' likelihood ratio test statistic is approximately of the form
where , and . What is the asymptotic distribution of this statistic?
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3.II.26I
Part II, 2007 comment(i) In the context of decision theory, what is a Bayes rule with respect to a given loss function and prior? What is an extended Bayes rule?
Characterise the Bayes rule with respect to a given prior in terms of the posterior distribution for the parameter given the observation. When for some , and the loss function is , what is the Bayes rule?
(ii) Suppose that , with loss function and suppose further that under .
Supposing that a prior is taken over , compute the Bayes risk of the decision rule . Find the posterior distribution of given , and confirm that its mean is of the form for some value of which you should identify. Hence show that the decision rule is an extended Bayes rule.
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4.II.27I
Part II, 2007 commentAssuming sufficient regularity conditions on the likelihood for a univariate parameter , establish the Cramér-Rao lower bound for the variance of an unbiased estimator of .
If is an unbiased estimator of whose variance attains the Cramér-Rao lower bound for every value of , show that the likelihood function is an exponential family.
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1.II.28J
Part II, 2007 comment(i) What does it mean to say that a process is a martingale? What does the martingale convergence theorem tell us when applied to positive martingales?
(ii) What does it mean to say that a process is a Brownian motion? Show that with probability one.
(iii) Suppose that is a Brownian motion. Find such that
is a martingale. Discuss the limiting behaviour of and for this as .
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2.II.28J
Part II, 2007 commentIn the context of a single-period financial market with traded assets, what is an arbitrage? What is an equivalent martingale measure?
Fix and consider the following single-period market with 3 assets:
Asset 1 is a riskless bond and pays no interest.
Asset 2 is a stock with initial price per share; its possible final prices are with probability and with probability .
Asset 3 is another stock that behaves like an independent copy of asset 2 .
Find all equivalent martingale measures for the problem and characterise all contingent claims that can be replicated.
Consider a contingent claim that pays 1 if both risky assets move in the same direction and zero otherwise. Show that the lower arbitrage bound, simply obtained by calculating all possible prices as the pricing measure ranges over all equivalent martingale measures, is zero. Why might someone pay for such a contract?
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3.II.27J
Part II, 2007 commentSuppose that over two periods a stock price moves on a binomial tree

(i) Determine for what values of the riskless rate there is no arbitrage. From here on, fix and determine the equivalent martingale measure.
(ii) Find the time-zero price and replicating portfolio for a European put option with strike 15 and expiry
(iii) Find the time-zero price and optimal exercise policy for an American put option with the same strike and expiry.
(iv) Deduce the corresponding (European and American) call option prices for the same strike and expiry.
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4.II.28J
Part II, 2007 commentBriefly describe the Black-Scholes model. Consider a "cash-or-nothing" option with strike price , i.e. an option whose payoff at maturity is
It can be interpreted as a bet that the stock will be worth at least at time . Find a formula for its value at time , in terms of the spot price . Find a formula for its Delta (i.e. its hedge ratio). How does the Delta behave as ? Why is it difficult, in practice, to hedge such an instrument?
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2.II.29I
Part II, 2007 commentState Pontryagin's maximum principle in the case where both the terminal time and the terminal state are given.
Show that is the minimum value taken by the integral
subject to the constraints and , where
[You may find it useful to note the fact that the problem is rotationally symmetric about the -axis, so that the angle made by the initial velocity with the positive -axis may be chosen arbitrarily.]
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3.II.28I
Part II, 2007 commentLet be a discrete-time controllable dynamical system (or Markov decision process) with countable state-space and action-space . Consider the -horizon dynamic optimization problem with instantaneous costs , on choosing action in state at time , with terminal cost , in state at time . Explain what is meant by a Markov control and how the choice of a control gives rise to a time-inhomogeneous Markov chain.
Suppose we can find a bounded function and a Markov control such that
with equality when , and such that for all . Here denotes the expected value of , given that we choose action in state at time . Show that is an optimal Markov control.
A well-shuffled pack of cards is placed face-down on the table. The cards are turned over one by one until none are left. Exactly once you may place a bet of on the event that the next two cards will be red. How should you choose the moment to bet? Justify your answer.
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4.II.29I
Part II, 2007 commentConsider the scalar controllable linear system, whose state evolves by
with observations given by
Here, is the control variable, which is to be determined on the basis of the observations up to time , and are independent random variables. You wish to minimize the long-run average expected cost, where the instantaneous cost at time is . You may assume that the optimal control in equilibrium has the form , where is given by a recursion of the form
and where is chosen so that is independent of the observations up to time . Show that , and determine the minimal long-run average expected cost. You are not expected to simplify the arithmetic form of your answer but should show clearly how you have obtained it.
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1.II.29A
Part II, 2007 comment(i) Consider the problem of solving the equation
for a function , with data specified on a hypersurface
Assume that are functions. Define the characteristic curves and explain what it means for the non-characteristic condition to hold at a point on . State a local existence and uniqueness theorem for the problem.
(ii) Consider the case and the equation
with data specified on the axis . Obtain a formula for the solution.
(iii) Consider next the case and the equation
with data specified on the hypersurface , which is given parametrically as where is defined by
Find the solution and show that it is a global solution. (Here "global" means is on all of .)
(iv) Consider next the equation
to be solved with the same data given on the same hypersurface as in (iii). Explain, with reference to the characteristic curves, why there is generally no global solution. Discuss the existence of local solutions defined in some neighbourhood of a given point for various . [You need not give formulae for the solutions.]
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2.II.30A
Part II, 2007 commentDefine (i) the Fourier transform of a tempered distribution , and (ii) the convolution of a tempered distribution and a Schwartz function . Give a formula for the Fourier transform of ("convolution theorem").
Let . Compute the Fourier transform of the tempered distribution defined by
and deduce the Kirchhoff formula for the solution of
Prove, by consideration of the quantities and , that any solution is also given by the Kirchhoff formula (uniqueness).
Prove a corresponding uniqueness statement for the initial value problem
where is a smooth positive real-valued function of only.
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3.II.29A
Part II, 2007 commentWrite down the formula for the solution for of the initial value problem for the heat equation in one space dimension
for a given smooth bounded function.
Define the distributional derivative of a tempered distribution . Define a fundamental solution of a constant-coefficient linear differential operator , and show that the distribution defined by the function is a fundamental solution for the operator .
For the equation
where , prove that there is a unique solution of the form with . Hence write down the solution of with general initial data and describe the large time behaviour.
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4.II.30A
Part II, 2007 commentState and prove the mean value property for harmonic functions on .
Obtain a generalization of the mean value property for sub-harmonic functions on , i.e. functions for which
for all .
Let solve the equation
where is a real-valued continuous function. By considering the function show that, on any ball ,
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1.II.30B
Part II, 2007 commentState Watson's lemma, describing the asymptotic behaviour of the integral
as , given that has the asymptotic expansion
as , where and .
Give an account of Laplace's method for finding asymptotic expansions of integrals of the form
for large real , where is real for real .
Deduce the following asymptotic expansion of the contour integral
as .
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3.II.30B
Part II, 2007 commentExplain the method of stationary phase for determining the behaviour of the integral
for large . Here, the function is real and differentiable, and and are all real.
Apply this method to show that the first term in the asymptotic behaviour of the function
where with and real, is
as
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4.II.31B
Part II, 2007 commentConsider the time-independent Schrödinger equation
where denotes and denotes . Suppose that
and consider a bound state . Write down the possible Liouville-Green approximate solutions for in each region, given that as .
Assume that may be approximated by near , where , and by near , where . The Airy function satisfies
and has the asymptotic expansions
and
Deduce that the energies of bound states are given approximately by the WKB condition:
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1.II.31E
Part II, 2007 comment(i) Using the Cole-Hopf transformation
map the Burgers equation
to the heat equation
(ii) Given that the solution of the heat equation on the infinite line with initial condition is given by
show that the solution of the analogous problem for the Burgers equation with initial condition is given by
where the function is to be determined in terms of .
(iii) Determine the ODE characterising the scaling reduction of the spherical modified Korteweg-de Vries equation
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2.II.31E
Part II, 2007 commentSolve the following linear singular equation
where denotes the unit circle, and denotes the principal value integral.
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3.II.31E
Part II, 2007 commentFind a Lax pair formulation for the linearised NLS equation
Use this Lax pair formulation to show that the initial value problem on the infinite line of the linearised NLS equation is associated with the following Riemann-Hilbert problem
By deforming the above problem obtain the Riemann-Hilbert problem and hence the linear integral equation associated with the following system of nonlinear evolution PDEs
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1.II.32D
Part II, 2007 commentA particle in one dimension has position and momentum operators and whose eigenstates obey
Given a state , define the corresponding position-space and momentum-space wavefunctions and and show how each of these can be expressed in terms of the other. Derive the form taken in momentum space by the time-independent Schrödinger equation
for a general potential .
Now let with a positive constant. Show that the Schrödinger equation can be written
and verify that it has a solution for unique choices of and , to be determined (you need not find the normalisation constant, ). Check that this momentum space wavefunction can also be obtained from the position space solution .
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2.II.32D
Part II, 2007 commentLet denote the combined spin eigenstates for a system of two particles, each with spin 1. Derive expressions for all states with in terms of product states.
Given that the particles are identical, and that the spatial wavefunction describing their relative position has definite orbital angular momentum , show that must be even. Suppose that this two-particle state is known to arise from the decay of a single particle, , also of spin 1. Assuming that total angular momentum and parity are conserved in this process, find the values of and that are allowed, depending on whether the intrinsic parity of is even or odd.
[You may set and use ]
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3.II.32D
Part II, 2007 commentLet
be the position and momentum operators for a one-dimensional harmonic oscillator of mass and frequency . Write down the commutation relations obeyed by and and give an expression for the oscillator Hamiltonian in terms of them. Prove that the only energies allowed are with and give, without proof, a formula for a general normalised eigenstate in terms of .
A three-dimensional oscillator with charge is subjected to a weak electric field so that its total Hamiltonian is
where for and is a small, dimensionless parameter. Express the general eigenstate for the Hamiltonian with in terms of one-dimensional oscillator states, and give the corresponding energy eigenvalue. Use perturbation theory to compute the changes in energies of states in the lowest two levels when , working to the leading order at which non-vanishing corrections occur.
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4.II.32D
Part II, 2007 commentThe Hamiltonian for a particle of spin in a magnetic field is
and is a constant (the motion of the particle in space can be ignored). Consider a magnetic field which is independent of time. Writing , where is a unit vector, calculate the time evolution operator and show that if the particle is initially in a state the probability of measuring it to be in an orthogonal state after a time is
Evaluate this to find the probability for a transition from a state of spin up along the direction to one of spin down along the direction when .
Now consider a magnetic field whose and components are time-dependent but small:
Show that the probability for a transition from a spin-up state at time zero to a spin-down state at time (with spin measured along the direction, as before) is approximately
where you may assume . Comment on how this compares, when , with the result for a time-independent field.
[The first-order transition amplitude due to a perturbation is
where and are orthogonal eigenstates of the unperturbed Hamiltonian with eigenvalues and respectively.]
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1.II.33A
Part II, 2007 commentIn a certain spherically symmetric potential, the radial wavefunction for particle scattering in the sector ( -wave), for wavenumber and , is
where
with and real, positive constants. Scattering in sectors with can be neglected. Deduce the formula for the -matrix in this case and show that it satisfies the expected symmetry and reality properties. Show that the phase shift is
What is the scattering length for this potential?
From the form of the radial wavefunction, deduce the energies of the bound states, if any, in this system. If you were given only the -matrix as a function of , and no other information, would you reach the same conclusion? Are there any resonances here?
[Hint: Recall that for real , where is the phase shift.]
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2.II.33A
Part II, 2007 commentDescribe the variational method for estimating the ground state energy of a quantum system. Prove that an error of order in the wavefunction leads to an error of order in the energy.
Explain how the variational method can be generalized to give an estimate of the energy of the first excited state of a quantum system.
Using the variational method, estimate the energy of the first excited state of the anharmonic oscillator with Hamiltonian
How might you improve your estimate?
[Hint: If then
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3.II.33A
Part II, 2007 commentConsider the Hamiltonian
for a particle of spin fixed in space, in a rotating magnetic field, where
and
with and constant, and .
There is an exact solution of the time-dependent Schrödinger equation for this Hamiltonian,
where and
Show that, for , this exact solution simplifies to a form consistent with the adiabatic approximation. Find the dynamic phase and the geometric phase in the adiabatic regime. What is the Berry phase for one complete cycle of ?
The Berry phase can be calculated as an integral of the form
Evaluate for the adiabatic evolution described above.
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4.II.33A
Part II, 2007 commentConsider a 1-dimensional chain of atoms of mass (with large and with periodic boundary conditions). The interactions between neighbouring atoms are modelled by springs with alternating spring constants and , with .

In equilibrium, the separation of the atoms is , the natural length of the springs.
Find the frequencies of the longitudinal modes of vibration for this system, and show that they are labelled by a wavenumber that is restricted to a Brillouin zone. Identify the acoustic and optical bands of the vibration spectrum, and determine approximations for the frequencies near the centre of the Brillouin zone. What is the frequency gap between the acoustic and optical bands at the zone boundary?
Describe briefly the properties of the phonons in this system.
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2.II.34D
Part II, 2007 commentDerive the Maxwell relation
The diagram below illustrates the Joule-Thomson throttling process for a porous barrier. A gas of volume , initially on the left-hand side of a thermally insulated pipe, is forced by a piston to go through the barrier using constant pressure . As a result the gas flows to the right-hand side, resisted by a piston which applies a constant pressure (with ). Eventually all of the gas occupies a volume on the right-hand side. Show that this process conserves enthalpy.
The Joule-Thomson coefficient is the change in temperature with respect to a change in pressure during a process that conserves enthalpy . Express the JouleThomson coefficient, , in terms of , the heat capacity at constant pressure , and the volume coefficient of expansion .
What is for an ideal gas?
If one wishes to use the Joule-Thomson process to cool a real (non-ideal) gas, what must the of be?

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3.II.34D
Part II, 2007 commentFor a 2-dimensional gas of nonrelativistic, non-interacting, spinless bosons, find the density of states in the neighbourhood of energy . [Hint: consider the gas in a box of size which has periodic boundary conditions. Work in the thermodynamic limit , with held finite.]
Calculate the number of particles per unit area at a given temperature and chemical potential.
Explain why Bose-Einstein condensation does not occur in this gas at any temperature.
[Recall that
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4.II.34D
Part II, 2007 commentConsider a classical gas of diatomic molecules whose orientation is fixed by a strong magnetic field. The molecules are not free to rotate, but they are free to vibrate. Assuming that the vibrations are approximately harmonic, calculate the contribution to the partition function due to vibrations.
Evaluate the free energy , where is the total partition function for the gas, and hence calculate the entropy.
Note that and You may approximate ! by .]
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1.II.34E
Part II, 2007 commentFrame is moving with uniform speed in the -direction relative to a laboratory frame . Using Cartesian coordinates and units such that , the relevant Lorentz transformation is
where . A straight thin wire of infinite extent lies along the -axis and carries charge and current line densities and per unit length, as measured in . Stating carefully your assumptions show that the corresponding quantities in are given by
Using cylindrical polar coordinates, and the integral forms of the Maxwell equations and , derive the electric and magnetic fields outside the wire in both frames.
In a standard notation the Lorentz transformation for the electric and magnetic fields is
Is your result consistent with this?
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3.II
Part II, 2007 commentConsider a particle of charge moving with 3 -velocity . If the particle is moving slowly then Larmor's formula asserts that the instantaneous radiated power is
Suppose, however, that the particle is moving relativistically. Give reasons why one should conclude that is a Lorentz invariant. Writing the 4-velocity as where and , show that
where and where is the particle's proper time. Show also that
Deduce the relativistic version of Larmor's formula.
Suppose the particle moves in a circular orbit perpendicular to a uniform magnetic field . Show that
where is the mass of the particle, and comment briefly on the slow motion limit.
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4.II.35E
Part II, 2007 commentAn action
is given, where is a scalar field. Explain heuristically how to compute the functional derivative .
Consider the action for electromagnetism,
Here is the 4-current density, is the 4-potential and is the Maxwell field tensor. Obtain Maxwell's equations in 4-vector form.
Another action that is sometimes suggested is
Under which additional assumption can Maxwell's equations be obtained using this action?
Using this additional assumption establish the relationship between the actions and .
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1.II.35A
Part II, 2007 commentStarting from the Riemann tensor for a metric , define the Ricci tensor and the scalar curvature .
The Riemann tensor obeys
Deduce that
Write down Einstein's field equations in the presence of a matter source, with energymomentum tensor . How is the relation important for the consistency of Einstein's equations?
Show that, for a scalar function , one has
Assume that
for a scalar field . Show that the quantity
is then a constant.
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2.II.35A
Part II, 2007 commentThe symbol denotes the covariant derivative defined by the Christoffel connection for a metric . Explain briefly why
in general, where is a scalar field and is a covariant vector field.
A Killing vector field satisfies the equation
By considering the quantity , show that
Find all Killing vector fields in the case of flat Minkowski space-time.
For a metric of the form
where denotes the coordinates , show that and that . Deduce that the vector field is a Killing vector field.
[You may assume the standard symmetries of the Riemann tensor.]
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4.II.36A
Part II, 2007 commentConsider a particle on a trajectory . Show that the geodesic equations, with affine parameter , coincide with the variational equations obtained by varying the integral
the end-points being fixed.
In the case that , show that the space-time metric is given in the form
for a certain function . Assuming the particle motion takes place in the plane show that
for constants. Writing , obtain the equation
where can be chosen to be 1 or 0 , according to whether the particle is massive or massless. In the case that , show that
In the massive case, show that there is an approximate solution of the form
where
What is the interpretation of this solution?
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1.II.36B
Part II, 2007 commentDiscuss how the methods of lubrication theory may be used to find viscous fluid flows in thin layers or narrow gaps, explaining carefully what inequalities need to hold in order that the theory may apply.
Viscous fluid of kinematic viscosity flows under the influence of gravity , down an inclined plane making an angle with the horizontal. The fluid layer lies between and , where are distances measured down the plane and perpendicular to it, and is of the same order as . Give conditions involving and that ensure that lubrication theory can be used, and solve the lubrication equations, together with the equation of mass conservation, to obtain an equation for in the form
where are constants to be determined. Show that there is a steady solution with constant, and interpret this physically. Show also that a solution of this equation exists in the form of a front, with , where , and as . Determine in terms of , find the shape of the front implicitly in the form , and show that as from below.
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2.II.36B
Part II, 2007 commentViscous fluid is extracted through a small hole in the tip of the cone given by in spherical polar coordinates . The total volume flux through the hole takes the constant value . It is given that there is a steady solution of the Navier-Stokes equations for the fluid velocity . For small enough , the velocity is well approximated by , where except in thin boundary layers near .
(i) Verify that the volume flux through the hole is approximately .
(ii) Construct a Reynolds number (depending on ) in terms of and the kinematic viscosity , and thus give an estimate of the value of below which solutions of this type will appear.
(iii) Assuming that there is a boundary layer near , write down the boundary layer equations in the usual form, using local Cartesian coordinates and parallel and perpendicular to the boundary. Show that the boundary layer thickness is proportional to , and show that the component of the velocity may be written in the form
Derive the equation and boundary conditions satisfied by . Give an expression, in terms of , for the volume flux through the boundary layer, and use this to derive the dependence of the first correction to the flow outside the boundary layer.
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3.II B
Part II, 2007 comment
Viscous fluid of kinematic viscosity and density flows in a curved pipe of constant rectangular cross section and constant curvature. The cross-section has height and width (in the radial direction) with , and the radius of curvature of the inner wall is , with . A uniform pressure gradient is applied along the pipe.
(i) Assume to a first approximation that the pipe is straight, and ignore variation in the -direction, where are Cartesian coordinates referred to an origin at the centre of the section, with increasing radially and measured along the pipe. Find the flow field along the pipe in the form .
(ii) It is given that the largest component of the inertial acceleration due to the curvature of the pipe is in the direction. Consider the secondary flow induced in the plane, again ignoring variations in and any end effects (except for the requirement that there be zero total mass flux in the direction). Show that takes the form , where
and write down two equations determining the constants and . [It is not necessary to solve these equations.]
Give conditions on the parameters that ensure that .
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4.II.37B
Part II, 2007 comment(i) Assuming that axisymmetric incompressible flow , with vorticity in spherical polar coordinates satisfies the equations
where
show that for Stokes flow satisfies the equation
(ii) A rigid sphere of radius moves at velocity through viscous fluid of density and dynamic viscosity which is at rest at infinity. Assuming Stokes flow and by applying the boundary conditions at and as , verify that is the appropriate solution to for this flow, where and are to be determined.
(iii) Hence find the velocity field outside the sphere. Without direct calculation, explain why the drag is in the direction and has magnitude proportional to .
(iv) A second identical sphere is introduced into the flow, at a distance from the first, and moving at the same velocity. Justify the assertion that, when the two spheres are at the same height, or when one is vertically above the other, the drag on each sphere is the same. Calculate the leading correction to the drag in each case, to leading order in .
[You may quote without proof the fact that, for an axisymmetric function ,
in spherical polar coordinates .]
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1.II.37C
Part II, 2007 commentA uniform elastic solid with density and Lamé moduli and occupies the region between rigid plane boundaries and . Show that SH waves can propagate in the direction within this layer, and find the dispersion relation for such waves.
Deduce for each mode (a) the cutoff frequency, (b) the phase velocity, and (c) the group velocity.
Show also that for each mode the kinetic energy and elastic energy are equal in an average sense to be made precise.
[You may assume that the elastic energy per unit volume .]
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2.II.37C
Part II, 2007 commentShow that for a one-dimensional flow of a perfect gas at constant entropy the Riemann invariants are constant along characteristics .
Define a simple wave. Show that in a right-propagating simple wave
Now suppose instead that, owing to dissipative effects,
where is a positive constant. Suppose also that is prescribed at for all , say . Demonstrate that, unless a shock forms,
where, for each and is determined implicitly as the solution of the equation
Deduce that a shock will not form at any if
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3.II.37C
Part II, 2007 commentWaves propagating in a slowly-varying medium satisfy the local dispersion relation
in the standard notation. Give a brief derivation of the ray-tracing equations for such waves; a formal justification is not required.
An ocean occupies the region . Water waves are incident on a beach near . The undisturbed water depth is
with a small positive constant and positive. The local dispersion relation is
and where are the wavenumber components in the directions. Far from the beach, the waves are planar with frequency and crests making an acute angle with the shoreline . Obtain a differential equation (in implicit form) for a ray , and show that near the shore the ray satisfies
where and should be found. Sketch the appearance of the wavecrests near the shoreline.
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4.II
Part II, 2007 commentShow that, for a plane acoustic wave, the acoustic intensity may be written as in the standard notation.
Derive the general spherically-symmetric solution of the wave equation. Use it to find the velocity potential for waves radiated into an unbounded fluid by a pulsating sphere of radius
By considering the far field, or otherwise, find the time-average rate at which energy is radiated by the sphere.
You may assume that .]
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1.II.38C
Part II, 2007 comment(a) For a numerical method to solve , define the linear stability domain and state when such a method is A-stable.
(b) Determine all values of the real parameter for which the Runge-Kutta method
is A-stable.
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2.II.38C
Part II, 2007 comment(a) State the Householder-John theorem and explain how it can be used to design iterative methods for solving a system of linear equations .
(b) Let where is the diagonal part of , and and are, respectively, the strictly lower and strictly upper triangular parts of . Given a vector , consider the following iterative scheme:
Prove that if is a symmetric positive definite matrix, and , then the above iteration converges to the solution of the system .
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3.II.38C
Part II, 2007 comment(a) Prove that all Toeplitz symmetric tridiagonal matrices
share the same eigenvectors with components , , and eigenvalues to be determined.
(b) The diffusion equation
is approximated by the Crank-Nicolson scheme
where , and is an approximation to . Assuming that show that the above scheme can be written in the form
where and the real matrices and should be found. Using matrix analysis, find the range of for which the scheme is stable. [Do not use Fourier analysis.]
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4.II.39C
Part II, 2007 comment(a) Suppose that is a real matrix, and that and are given so that . Further, let be a non-singular matrix such that , where is the first coordinate vector and . Let . Prove that the eigenvalues of are together with the eigenvalues of the bottom right submatrix of
(b) Suppose again that is a real matrix, and that two linearly independent vectors are given such that the linear subspace spanned by and is invariant under the action of , i.e.,
Denote by an matrix whose two columns are the vectors and , and let be a non-singular matrix such that is upper triangular, that is,
Again let . Prove that the eigenvalues of are the eigenvalues of the top left submatrix of together with the eigenvalues of the bottom right submatrix of